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Galileo’s Experiments with Pendulums: Then and Now

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Galileo’s Experiments with Pendulums: Then and Now

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For small oscillations, a pendulum moves with armonic motion.
The period of a simple pendulum depends on:

the length of the non-extensible string

the acceleration of gravity g

The period of a simple pendulum does not depend on:

the mass m attached to the string

the amplitude of the pendulum's swing (isochronism of small oscillations)

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If a point P moves in a uniform circular motion, the motion of its projection on the diameter of the circle is a harmonic motion.
In  harmonic motion:

the speed at the ends of oscillation is zero and the maximum speed occurs in the middle of the oscillation

the magnitude of the acceleration is greatest at the ends of the oscillation and the acceleration is zero in the middle of the oscillation

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From the first scientific investigations of the pendulum around 1602 by Galileo Galilei, the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s.

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A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth.
The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing.
From the first scientific investigations of the pendulum around 1602 by Galileo Galilei, the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s.
The pendulum clock invented by Christian Huygens in 1658 became the world's standard timekeeper, used in homes and offices for 270 years, and achieved accuracy of about one second per year before it was superseded as a time standard by quartz clocks in the 1930s. Pendulums are also used in scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the acceleration of gravity in geophysical surveys, and even as a standard of length.
The word "pendulum" is new Latin, from the Latin pendulus, meaning 'hanging'.
https://en.wikipedia.org/wiki/Pendulum

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The motion of an undamped pendulum approximates to simple harmonic motion if the angle of oscillation is small.

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Simple harmonic motion can serve as a mathematical model for a variety of motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum as well as molecular vibration.

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In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a pendulum of length l with gravitational acceleration g is given by

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Simple harmonic motion can be considered the one-dimensional projection of uniform circular motion. If an object moves with angular speed ω around a circle of radius r centered at the origin of the xy-plane, then its motion along each coordinate is simple harmonic motion with amplitude r and angular frequency ω.

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Simple harmonic motion can serve as a mathematical model for a variety of motions, such as the oscillation of a spring. In addition, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum as well as molecular vibration.

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How can the motion of an object in a circle at a constant speed be described? Is such an object accelerating or not? If there is an acceleration, then what direction is it and why does it not change the object's speed? How does a change in mass, speed, or radius of the circle affect quantities such as the acceleration and net force? Explore these questions and build the foundation for further studies with the Uniform Circular Motion Interactive.

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The vibration of a guitar string is an example of simple harmonic motion.

Credit: Joshua David Treisner | Shutterstock

When a musician strums a guitar, the vibration of the strings creates sound waves that human ears hear as music. When a guitar string is plucked, it moves a certain distance, depending on how hard the guitar player strums. The string returns to its starting point and travels nearly the same distance in the opposite direction. The vibrational energy of the string is dissipated in the form of sound. This causes the distance the string moves, or the amplitude of the vibrations, to decrease gradually. The volume of the sound fades until the string eventually falls silent.

The guitar string is an example of simple harmonic motion, or SHM. SHM can be seen throughout nature. It describes the vibration of atoms, the variability of giant stars, and countless other systems from musical instruments to swaying skyscrapers.

Princeton University's WordNet defines simple harmonic motion as "periodic motion in which the restoring force is proportional to the displacement." Periodic means that the motion repeats at a constant, predictable rate. The part about the restoring force being proportional to the displacement simply means the farther you push something, the harder it pushes back.

SHM also describes the motion of a ball hanging from a spring. The ball is subject to a linear elastic restoring force, according to Georgia State University's HyperPhysics website. Drawn out on a graph, the up-and-down motion of the ball over time can be illustrated as a smooth, repetitive oscillation, or sinusoidal wave. The shape of the wave can be calculated using Hooke's Law.

Robert Hooke was a British scientist, mathematician and architect who was interested in many aspects of science and technology, from astronomy to microbiology. Studying objects under a magnifying glass, he was the first to use the word "cell" to describe the components of plant tissues, according to the Physics Hypertextbook. He also studied clocks and timekeeping, and in 1675 developed a theory of elasticity or springiness as a way to regulate pendulum clocks. In short, the theory says, "Extension is directly proportional to force."

Mathematically, Hooke's Law is expressed as F = − kx, where F is the force, x is the change in length from the spring's relaxed or original length, and k is the characteristic spring constant that specifies the amount of force required to stretch or compress the spring a certain distance. The minus sign indicates that the restoring force is in the opposite direction of the displacement.

A ball on a spring is the standard example of periodic motion. If the displacement of the mass is plotted as a function of time, it will trace out a sinusoidal wave.

Credit: Georgia State University

There is a close connection between circular motion and simple harmonic motion, according to Boston University. Consider a point on the rim of a disk as it rotates counterclockwise at a constant rate around a horizontal axis. If we plot only the vertical position of the point as the disk turns, it produces a sinusoidal graph. This is exactly the same graph as we get if we plot the position of a mass on a spring bouncing up and down in simple harmonic motion as a function of time.

Simple pendulums behave much like harmonic oscillators such as springs. However, the period of a pendulum is determined not by its mass but by its length. Another significant difference is that in the case of a pendulum, the restoring force is provided not by a spring but by gravity. Since gravity is pulling the pendulum vertically downward and not back along the arc in the opposite direction of its motion, the restoring force is a somewhat complex trigonometric function. Nevertheless, pendulums closely approximate simple harmonic motion provided they don't swing more than a few degrees from their resting point.

All mechanical systems are subject to damping forces, which cause the amplitude of the motion to decrease over time. These forces can include frictional forces between moving parts, air resistance or internal forces such as those in springs that tend to dissipate energy as heat.

In simple harmonic motion, the damping force is generally proportional to the velocity of the oscillating mass, according to the State University of New York Stony Brook. This force will eventually bring an oscillating system to a halt, and if it is great enough, it can actually stop oscillations before they can start.

A weight on a spring bouncing in air will continue bouncing for quite a long time, but not forever. Air resistance and internal resistance in the metal spring will eventually dissipate the kinetic energy of the system and bring it to a stop. However, if the weight were suspended in a pot of cold molasses, and the spring is stretched and released, the weight might only return slowly to its rest position without bouncing above it even once. We consider the weight bouncing in air to be underdamped, and the weight moving slowly through molasses and never bouncing even once to be overdamped. If the system is exactly on the cusp between these two conditions, it is said to be critically damped.

"We know that in reality, a spring won't oscillate forever. Frictional forces will diminish the amplitude of oscillation until eventually the system is at rest," according to the University of California Santa Cruz. The amplitude of an underdamped oscillator undergoes an exponential decay, meaning that after a certain time, the amplitude of the oscillations will decrease by half, and after that same time period, it will decrease again by half. One familiar example of this is the decaying sound of a bell.

In many cases, damping is desired to prevent uncontrolled bouncing. Perhaps the most familiar example of this is an automobile suspension system. The wheels are mounted on springs so they can move up and down in response to bumps and dips in the road while the body of the car remains relatively level. However, without damping, the car would begin to bounce up and down uncontrollably. To prevent this, cars have shock absorbers to damp the movement of the springs by adding a frictional force to the suspension system.

When it is desirable for harmonic motion to persist, damping is a problem that must be overcome with a driving force. Consider the pendulum in a grandfather clock. At its farthest point in each direction, an escapement mechanism, powered by the gravitational energy of a slowly descending weight, gives the pendulum a tiny nudge that is sufficient to overcome the damping effect of mechanical friction and air resistance and maintain the pendulum's motion.

One manifestation of driven harmonic motion is coupled resonance, or sympathetic vibration. This is a vibration that is produced in one body by the vibrations of exactly the same period in a neighboring body. One familiar example of this is a playground swing set. If it has two swings of the same length, swinging on one swing can cause the other one to start moving. The reason for this is that the first swing causes the supporting cross bar to move forward and back slightly with each cycle. Although this motion is usually small, because swings are the same length, they will have the same resonant frequency, so the driving force provided by the tiny motions of the cross bar become amplified, causing the second swing to move noticeably. Another example of a harmonic oscillator being driven at its resonant frequency is how an electric guitar can sustain a note indefinitely by allowing feedback of the amplified sound to drive the continued vibration of the string.

Sometimes, though, harmonic motion under a driving force can be harmful or even disastrous. If this force is applied at the natural or resonant frequency of the oscillating system at a rate that is greater than damping forces can dissipate the energy, "...relatively small driving forces can build up to large-amplitude oscillations, just because energy is continually being injected into the system at just the right frequency," according to Boston University. This means that if an underdamped system is driven by external forces at its resonant frequency, the amplitude of the motion can increase until the system fails. One of the most dramatic examples of this is the 1940 Tacoma Narrows Bridge collapse. Strong sustained winds drove oscillations of the bridge deck that increased in amplitude until it broke apart. Another example of harmonic motion being driven to the point of failure is how singer can shatter a wine glass by loudly singing a note at its resonant frequency.

Harmonic motion in real life is rarely simple. When we look at real macroscopic oscillating systems, there can be any number confounding variables, such as fluctuations in temperature and air pressure or wear and tear on mechanical parts, which can affect the amplitude and period of the motion. However, by assuming that mechanical systems are in simple harmonic motion, we can often make suitably accurate approximations of how these systems behave.

Additional resources

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When a musician strums a guitar, the vibration of the strings creates sound waves that human ears hear as music. When a guitar string is plucked, it moves a certain distance, depending on how hard the guitar player strums. The string returns to its starting point and travels nearly the same distance in the opposite direction.

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Motion that recurs over and over and the period of time required for each recurrence remains the same

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A periodic vibration, as of a pendulum, in which the motions are symmetrical about a region of equilibrium. Such avibration may have only one frequency and amplitude or may be a combination of two or more components. Alsocalled periodic motion, simple harmonic motion.

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There are two basic ways to measure time: by duration or periodic motion. Early clocks measured duration by calibrating the burning of incense or wax, or the flow of water or sand from a container. Our calendar consists of years determined by the motion of the sun; months determined by the motion of the moon; days by the rotation of the earth; hours by the motion of cyclic motion of gear trains; and seconds by the oscillations of springs or pendulums. In modern times a second is defined by a specific number of vibrations of radiation, corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. Sundials calibrate the motion of the sun through the sky, including seasonal corrections. A clock escapement is a device that can transform continuous movement into discrete movements of a gear train. The early escapements used oscillatory motion to stop and start the turning of a weight-driven rotating drum. Soon, complicated escapements were regulated by pendulums, the theory of which was first developed by the physicist Christian Huygens in the mid 17th century. The accuracy of clocks was increased and the size reduced by the discovery of the oscillatory properties of springs by Robert Hooke. By the middle of the 18th century, the technology of timekeeping advanced to the point that William Harrison developed timekeeping devices that were accurate to one second in a century.
http://web.mit.edu/8.01t/www/materials/modules/chapter23.pdf

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Unit 4 Multiple Choice Test (Chapters 5 and 8) ~Please enter the letter of the best choice in the space available.

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Italiano

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What is centripetal acceleration? (article) | Khan Academy

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Learn what centripetal acceleration means and how to calculate it.

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Race cars with constant speed around curve

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Circular Motion

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Uploaded by Matt Anderson on 2014-02-23.

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Circular Motion

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Circular Motion | A-Level Physics | Doodle Science

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In practical life, the motions of objects need not necessarily be along a straight line always. As a matter of fact objects move curvilinear moreoften than in linear direction. For eminent clarity we will consider a motion around a circle. Any curvilinear motion can be attributed to a part of circular motion and hence the concept remains the same.
An object under a circular motion experiences different types of forces and in the process different accelerations, namely,
http://physics.tutorvista.com/motion/acceleration.html

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The wheel of bullock cart of radius 1m is moving with the speed of 5m per second.
Calculate its Angular speed.

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... from a lesson about Uniform circular motion

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... from a lesson about Uniform circular motion

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When an object is experiencing uniform circular motion, it is traveling in a circular path at a constant speed.
If r is the radius of the path, and we define the period, T, as the time it takes to make a complete circle, then the speed is given by the circumference over the period.
A similar equation relates the magnitude of the acceleration to the speed: These two equations can be combined to give the equation:

This is known as the centripetal acceleration; v2 / r is the special form the acceleration takes when we're dealing with objects experiencing uniform circular motion.
http://physics.bu.edu/~duffy/py105/Circular.html

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Angular speed = angular displacement per unit time. It's measured in rad/s.

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Although the magnitude of (tangential) velocity (= speed) is constant, its direction is constantly changing.  Since the velocity changes, an object moving along a circle at constant speed is accelerating, (because its velocity is changing).
This acceleration is called centripetal acceleration.

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Period T = time required to the object to complete one revolution. It's measured in seconds.

Frequency  = the ratio of the number of the revolutions to the time to complete them. It's measured in 1/s.

Frequency and period are inversely proportional.

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Italiano

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Italiano

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Movement of an object along a circular path at a constant speed. Since the body describes circular motion, its distance from the axis of rotation remains constant at all times.
Though the body's speed is constant, its velocity is not constant: velocity, a vector quantity, depends on both the body's speed and its direction of travel.
This changing velocity indicates the presence of an acceleration; this centripetal acceleration is of constant magnitude and directed at all times towards the axis of rotation. This acceleration is, in turn, produced by a centripetal force which is also constant in magnitude and directed towards the axis of rotation.

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an artificial satellite orbiting the Earth at constant height

a stone which is tied to a rope and is being swung in circles

a car turning through a curve in a race track

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Circular motion is a movement of an object along a circular path

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Cricket the WonderDog wanders away from her house at a constant velocity of 1 m/s, stopping only when she's 5m away (which, of course, takes 5 seconds). She then decides to take a short five-second rest in the grass. After her five second rest, she hears the dinner bell ring, so she runs back to the house at a speed of 2 m/s. The displacement-time graph for her motion would look something like this:
http://www.aplusphysics.com/courses/regents/kinematics/regents_motion_graphs.html

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Graphs and diagrams are terrific tools for understanding physics, and they are especially helpful for studying motion, a phenomenon that we are used to perceiving visually. We'll explore motion through the study of particle diagrams, displacement-time graphs, velocity-time graphs, and acceleration-time graphs.

Particle Diagrams

Particle Diagrams, sometimes referred to as ticker-tape diagrams or dot diagrams, show the position or displacement of an object at evenly spaced time intervals. Think of a particle diagram like an oil drip pattern... if your car has a steady oil drip, where one drop of oil falls to the ground every second, the pattern of the oil droplets on the ground could represent the motion of your car with respect to time.

By examining the oil drop pattern, a bystander could draw conclusions about the displacement, velocity, and acceleration of your car, even if they weren't able to watch your car drive by! The oil drop pattern is known as a particle, or ticker-tape, diagram.

From the particle diagram we can see that your car was moving either to the right or the left, and since the drops are evenly spaced, we can say with certainty that your car was moving at a constant velocity, and since velocity isn't changing, acceleration must be 0.

So what would the particle diagram look like if your car was accelerating to the right? Let's take a look and see!

The oil drops start close together on the left, and get further and further apart as the object moves toward the right. Of course, this pattern could also have been produced by a car moving from right to left, beginning with a high velocity at the right and slowing down as it moves toward the left. Because the velocity vector (pointing to the left) and the acceleration vector (pointing to the right) are in opposite directions, the object slows down. This is a case where, if you called to the right the positive direction, the car would have a negative velocity, a positive acceleration, and it would be slowing down. Check out the resulting particle diagram below!

Thought Question: Can you think of a case in which the car could have a negative velocity and a negative acceleration, yet be speeding up? Draw that case!

Displacement-Time Graphs

As you've observed, particle diagrams can help you understand an object's motion, but they don't always tell you the whole story. We'll have to investigate some other types of motion graphs to get a clearer picture.

The displacement time graph (also known as a d-t graph or position-time graph) shows the displacement (or, in the case of scalar quantities, distance) of an object as a function of time. Positive displacements indicate the object's position is in the positive direction from its starting point, while negative displacements indicate the object's position is opposite the positive direction. Let's look at a few examples.

Suppose Cricket the WonderDog wanders away from her house at a constant velocity of 1 m/s, stopping only when she's 5m away (which, of course, takes 5 seconds). She then decides to take a short five-second rest in the grass. After her five second rest, she hears the dinner bell ring, so she runs back to the house at a speed of 2 m/s. The displacement-time graph for her motion would look something like this:

As you can see from the plot, Cricket's displacement begins at zero meters at time zero. Then, as time progresses, Cricket's displacement increases at a rate of 1 m/s, so that after one second, Cricket is one meter away from her starting point. After two seconds, she's two meters away, and so forth, until she reaches her maximum displacement of five meters from her starting point at a time of five seconds. Cricket then remains at that position for 5 seconds while she takes a rest. Following her rest, at time t=10 seconds, Cricket hears the dinner bell and races back to the house at a speed of 2 m/s, so the graph ends when Cricket returns to her starting point at the house, a total distance traveled of 10m, and a total displacement of zero meters.

As we look at the d-t graph, notice that at the beginning, when Cricket is moving in a positive direction, the graph has a positive slope. When the graph is flat (has a zero slope) Cricket is not moving. And when the graph has a negative slope, Cricket is moving in the negative direction. It's also easy to see that the steeper the slope of the graph, the faster Cricket is moving.

Velocity-Time Graphs

Just as important to understanding motion is the velocity-time graph, which shows the velocity of an object on the y-axis, and time on the x-axis. Positive values indicate velocities in the positive direction, while negative values indicate velocities in the opposite direction. In reading these graphs, it's important to realize that a straight horizontal line indicates the object maintaining a constant velocity - it can still be moving, it's velocity just isn't changing. A value of 0 on the v-t graph indicates the object has come to a stop. If the graph crosses the x-axis, the object was moving in one direction, came to a stop, and switched the direction of its motion. Let's look at the v-t graph for Cricket the Wonderdog's Adventure from the d-t graph section:

For the first five seconds of Cricket's journey, we can see she maintains a constant velocity of 1 m/s. Then, when she stops to rest, her velocity changes to zero for the duration of her rest. Finally, when she races back to the house for dinner, she maintains a negative velocity of 2 m/s. Because velocity is a vector, the negative sign indicates that Cricket's velocity is in the opposite direction (we initially defined the direction away from the house as positive, so back toward the house must be negative!)

As I'm sure you can imagine, the d-t graph of an object's motion and the v-t graph of an object's motion are closely related. We'll explore these relationships in the next section.

Graph Transformations

In looking at a d-t graph, the faster an object's displacement changes, the steeper the slope of the line. Since velocity is the rate at which an object's displacement changes, the slope of the d-t graph at any given point in time gives you the velocity at that point in time. We can obtain the slope of the d-t graph using the following formula:

Realizing that the rise in our graph is actually ∆v, and the run is ∆t, we can substitute these variables into our slope equation to find:

With a little bit of interpretation, it's easy to show that our slope is really just displacement over time, which is the definition of velocity. Put directly, the slope of the d-t graph is the velocity.

Of course, it only makes sense that if you can determine velocity from the d-t graph, you should be able to work backward to determine displacement from the v-t graph. If you have a v-t graph, and you want to know how much an object's displacement has changed in a time interval, take the area under the curve within that time interval.

So, if taking the slope of the d-t graph gives you the rate of change of displacement, which we call velocity, what do you get when you take the slope of the v-t graph? You get the rate of change of velocity, which we call acceleration! The slope of the v-t graph, then, gives you acceleration:

Let's take a look at a sample problem from the June 2009 Regents Physics Exam:

Now that you've seen how to solve these types of problems, why don't you try a few on your own?

Acceleration-Time Graphs

Much like we did with velocity, we can make a plot of acceleration vs. time by plotting the rate of change of an object's velocity (its acceleration) on the y-axis, and placing time on the x-axis. For the purposes of the NY Regents Physics Course, we'll always deal with a constant acceleration - all of our graphs will be a straight horizontal line, either at a positive value, a negative value, or at 0 (indicating a constant velocity). It's important to understand, however, that in real life not all accelerations are constant.

When we took the slope of the d-t graph, we obtained an object's velocity. In the same way, taking the slope of the v-t graph gives you an object's acceleration. Going the other direction, when we analyzed the v-t graphs, we found that taking the area under the v-t graph provided us with information about the object's change in displacement. In similar fashion, taking the area under the a-t graph tells us how much an object's velocity changes.

Putting it all together, we can go from displacement-time to velocity-time by taking the slope, and we can go from velocity-time to acceleration-time by taking the slope. Or, going the other direction, the area under the acceleration-time curve gives you an object's change in velocity, and the area under the velocity-time curve gives you an object's change in displacement.

Let's take a look at an example problem to demonstrate this concept.

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Graphs and diagrams are terrific tools for understanding physics, and they are especially helpful for studying motion, a phenomenon that we are used to perceiving visually. We'll explore motion through the study of particle diagrams, displacement-time graphs, velocity-time graphs, and acceleration-time graphs.

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Position vs. time graphs (video) | Khan Academy

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How to read a position vs. time graph. Using the graph to determine displacement, distance, average velocity, average speed, instantaneous velocity, and instantaneous speed.

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Linear Motion - Physics Video by Brightstorm

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Time-saving physics video on linear motion. Linear motion is the motion that is natural to an object, a straight line. Newton's first law of motion states that an object in motion will continue in a straight line unless disturbed by a force.

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The motion of any object is only meaningful when given relative to something else. Relativity in motion assures us that the laws of Physics don't vary depending on how much inertia an object has. This is important because everything has some amount of inertia. Standing on the surface of the earth, we feel that we are not in motion, but because the earth is moving, so are we. If we measure the speed of a ball thrown atop a train, we can either measure the speed of the ball with respect to the train s motion or with respect to the motion of the Earth.

Today we're going to talk about relativity in motion. First for you what is my present speed right now where I stand? Go ahead and think about that, I'll give you some choices is it zero meters per second? Is it 30,000 meters per second? Or is it over 50,000 meters per second? Well if you said zero meters per second, you're correct that is my present speed given the relativity to the floor, where I stand relative to the floor I'm moving at zero meters per second. If you said 30,000 meters per second that is also correct, that is my speed relative to the sun as we stand on earth, we are moving at 30,000 meters per second around the sun. If you said over 50,000 meters per second that is also correct that is our estimated speed and people don't even really know the actual value but that is the distance that we're moving away from the center of the universe. So my speed is relative I could, dependent on what we're talking about and what we're relative to.

Okay let's look at some examples, I've got a race car, that race car is traveling at 300 kilometers per hour okay that speed is relative to the track. If that race car is passing another race car, going 295 kilometers per hour the speed relative to the slower race car is 5 kilometers per hour so again relativity. Let's look at another example, let's say I'm on a train, and that train is traveling at 10 meters per second, I am on top of that train doing a very unsafe thing, I'm throwing a football in the direction that the train is moving. Don't try this at home okay I'm throwing that at 8 meters per second. Okay, well the ball relative to the train is moving at 8 meters per second but the ball relative to the ground or people standing on the ground watching is 10+8, 18 meters per second okay. So that makes sense and this works really well at relatively low speeds and trains and cars and me we're all relatively slow.

Okay it doesn't work so well at really high speeds, so now let's look at another example. I've got a train moving at 10 meters per second and I have a head light on that train and that head light is traveling at the speed of light. Okay the speed of light is 300 million meters per second okay so I say well that's easy I've got 10 meters per second plus 300 million meters per second which is going to be 300 million and 10 meters per second okay. Unfortunately it doesn't work that way, Einstein discovered in what we call "Einstein's special relativity in motion" that nothing can go faster than the speed of light. So that light coming out of the front of the train no matter how fast that train is going, if it's going half the speed of light or if it's going 200 kilometers an hour, that light is traveling at the same speed. So these are the ways we work with relativity in motion.

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Relativity in Motion - Physics Video by Brightstorm

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Physics video explaining relativity in motion. Everything has some degree of motion but when we judge motion, we need to judge it next to the standard of complete stillness. When we talk about the motion of a ball we assume that the earth is standing still when, in fact, it is not.

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This article was most recently revised and updated by

acceleration, rate at which velocity changes with time, in terms of both speed and direction. A point or an object moving in a straight line is accelerated if it speeds up or slows down. Motion on a circle is accelerated even if the speed is constant, because the direction is continually changing. For all other kinds of motion, both effects contribute to the acceleration.

Because acceleration has both a magnitude and a direction, it is a vector quantity. Velocity is also a vector quantity. Acceleration is defined as the change in the velocity vector in a time interval, divided by the time interval. Instantaneous acceleration (at a precise moment and location) is given by the limit of the ratio of the change in velocity during a given time interval to the time interval as the time interval goes to zero ( see analysis: Instantaneous rates of change). For example, if velocity is expressed in metres per second, acceleration will be expressed in metres per second per second.

The Editors of Encyclopaedia Britannica Erik Gregersen.

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Acceleration | Definition, Facts, & Units

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Acceleration, rate at which velocity changes with time, in terms of both speed and direction. A point or an object moving in a straight line is accelerated if it speeds up or slows down. Motion on a circle is accelerated even if the speed is constant, because the direction is continually changing.

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Velocity: Velocity, quantity that designates how fast and in what direction a point is moving. A point always moves in a direction that is tangent to its path; for a circular path, for example, its direction at any instant is perpendicular to a line from the point to the centre of the circle (a radius).

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Linear motion: Linear motion, motion in one spatial dimension. According to Newton's first law (also known as the principle of inertia), a body with no net force acting on it will either remain at rest or continue to move with uniform speed in a straight line, according to its initial condition of motion.

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Let's think about what physics is and what topics are covered in an introductory physics course.

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Fill the gap

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The slope of a velocity-time graph represents the acceleration of the object.

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Velocity-time graph

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Why distance is area under velocity-time line

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Speed is the magnitude of velocity. Velocity is a vector, which means it has two parts: first, your velocity has a magnitude, which just answers the question "how fast?", but does not say anything about the direction the object is moving.
The second part of velocity is its direction, which answers the question "which way?".
So a velocity might be "20 m/s, downward"; it means that:
the speed is 20 m/s and the direction is "downward".
Acceleration is the rate of change of velocity. Usually, acceleration means the speed is changing, but not always. When an object moves in a circular path at a constant speed, it is still accelerating, because the direction of its velocity is changing.

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If the acceleration has the same sign as the velocity, the object will be speeding up.
If the acceleration has the opposite sign as the velocity, the object will be slowing down.

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The formula that lets you find the final velocity after a time of constant acceleration:

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Acceleration is defined to be
the rate of change of the velocity

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If you’re not changing your speed and you’re not changing your direction, then you simply cannot be accelerating—no matter how fast you’re going. So, a jet moving with a constant velocity at 800 miles per hour along a straight line has zero acceleration, even though the jet is moving really fast, since the velocity isn’t changing. When the jet lands and quickly comes to a stop, it will have acceleration since it’s slowing down.

Or, you can think about it this way. In a car you could accelerate by hitting the gas or the brakes, either of which would cause a change in speed. But you could also use the steering wheel to turn, which would change your direction of motion. Any of these would be considered an acceleration since they change velocity.

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Compared to displacement and velocity, acceleration is like the angry, fire-breathing dragon of motion variables. It can be violent; some people are scared of it; and if it's big, it forces you to take notice. That feeling you get when you're sitting in a plane during take-off, or slamming on the brakes in a car, or turning a corner at a high speed in a go kart are all situations where you are accelerating.

Acceleration is the name we give to any process where the velocity changes. Since velocity is a speed and a direction, there are only two ways  for you to accelerate: change your speed or change your direction—or change both.

https://www.khanacademy.org/science/physics/one-dimensional-motion/acceleration-tutorial/a/acceleration-article

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Italiano

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Classical And Modern Physics, Divisions Of Physics....

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Physics - Classical And Modern Physics, Divisions Of Physics, Interrelationship Of Physics To Other Sciences, Physics And Philosophy

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An axiom among physicists-since the writings of Italian astronomer and physicist Galileo Galilei (1564-1642)-provides that the road to sure knowledge about the natural world is to carry out controlled observations (experiments) that will lead to measurable quantities. It is for this reason that experimental techniques, systems of measurements, and mathematical systems for expressing results lie at the core of research in physics.

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The study of motion by Galileo Galilei (1564-1642) in the late sixteenth and early seventeenth centuries and by Sir Isaac Newton in the mid-seventeenth century was one of the major cornerstones of modern Western experimental science.
Over a period of 20 years, Galileo observed the motions of objects rolling down various inclines and attempted to time these events. He discovered that the distance an object traveled was proportional to the square of the time that it was in motion. From these experiments came the first correct concept of accelerated motion.
Newton wanted to know why acceleration occurred. In order to produce a model that would help explain how the known universe of the seventeenth century worked, Newton had to give to science and physics the concept of a force which was mostly unknown at that time. With his second law of motion, he clearly demonstrated that acceleration is caused by an unbalanced force (commonly called a push or a pull) acting on an object. What we call gravity, Newton showed was nothing more than a special type of acceleration. The interaction of the acceleration of gravity on the mass of our body produces the force which is called weight. A general definition of mass is that it refers to the quantity of matter in a body.
http://science.jrank.org/pages/8/Acceleration-History.html

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Physical processes | MCAT | Test prep | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Just as distance and displacement have distinctly different meanings (despite their similarities), so do speed and velocity. Speed is a scalar quantity that refers to "how fast an object is moving." Speed can be thought of as the rate at which an object covers distance. A fast-moving object has a high speed and covers a relatively large distance in a short amount of time. Contrast this to a slow-moving object that has a low speed; it covers a relatively small amount of distance in the same amount of time. An object with no movement at all has a zero speed.

Velocity as a Vector Quantity

Velocity is a vector quantity that refers to "the rate at which an object changes its position." Imagine a person moving rapidly - one step forward and one step back - always returning to the original starting position. While this might result in a frenzy of activity, it would result in a zero velocity. Because the person always returns to the original position, the motion would never result in a change in position. Since velocity is defined as the rate at which the position changes, this motion results in zero velocity. If a person in motion wishes to maximize their velocity, then that person must make every effort to maximize the amount that they are displaced from their original position. Every step must go into moving that person further from where he or she started. For certain, the person should never change directions and begin to return to the starting position.

Velocity is a vector quantity. As such, velocity is direction aware. When evaluating the velocity of an object, one must keep track of direction. It would not be enough to say that an object has a velocity of 55 mi/hr. One must include direction information in order to fully describe the velocity of the object. For instance, you must describe an object's velocity as being 55 mi/hr, east. This is one of the essential differences between speed and velocity. Speed is a scalar quantity and does not keep track of direction; velocity is a vector quantity and is direction aware.

Determining the Direction of the Velocity Vector

The task of describing the direction of the velocity vector is easy. The direction of the velocity vector is simply the same as the direction that an object is moving. It would not matter whether the object is speeding up or slowing down. If an object is moving rightwards, then its velocity is described as being rightwards. If an object is moving downwards, then its velocity is described as being downwards. So an airplane moving towards the west with a speed of 300 mi/hr has a velocity of 300 mi/hr, west. Note that speed has no direction (it is a scalar) and the velocity at any instant is simply the speed value with a direction.

Calculating Average Speed and Average Velocity
As an object moves, it often undergoes changes in speed. For example, during an average trip to school, there are many changes in speed. Rather than the speed-o-meter maintaining a steady reading, the needle constantly moves up and down to reflect the stopping and starting and the accelerating and decelerating. One instant, the car may be moving at 50 mi/hr and another instant, it might be stopped (i.e., 0 mi/hr). Yet during the trip to school the person might average 32 mi/hr. The average speed during an entire motion can be thought of as the average of all speedometer readings. If the speedometer readings could be collected at 1-second intervals (or 0.1-second intervals or ... ) and then averaged together, the average speed could be determined. Now that would be a lot of work. And fortunately, there is a shortcut. Read on.

The average speed during the course of a motion is often computed using the following formula:

In contrast, the average velocity is often computed using this formula

Let's begin implementing our understanding of these formulas with the following problem:
Q: While on vacation, Lisa Carr traveled a total distance of 440 miles. Her trip took 8 hours. What was her average speed?
To compute her average speed, we simply divide the distance of travel by the time of travel.

That was easy! Lisa Carr averaged a speed of 55 miles per hour. She may not have been traveling at a constant speed of 55 mi/hr. She undoubtedly, was stopped at some instant in time (perhaps for a bathroom break or for lunch) and she probably was going 65 mi/hr at other instants in time. Yet, she averaged a speed of 55 miles per hour. The above formula represents a shortcut method of determining the average speed of an object.

Average Speed versus Instantaneous Speed

Since a moving object often changes its speed during its motion, it is common to distinguish between the average speed and the instantaneous speed. The distinction is as follows.

Instantaneous Speed - the speed at any given instant in time.

Average Speed - the average of all instantaneous speeds; found simply by a distance/time ratio.

You might think of the instantaneous speed as the speed that the speedometer reads at any given instant in time and the average speed as the average of all the speedometer readings during the course of the trip. Since the task of averaging speedometer readings would be quite complicated (and maybe even dangerous), the average speed is more commonly calculated as the distance/time ratio.

Moving objects don't always travel with erratic and changing speeds. Occasionally, an object will move at a steady rate with a constant speed. That is, the object will cover the same distance every regular interval of time. For instance, a cross-country runner might be running with a constant speed of 6 m/s in a straight line for several minutes. If her speed is constant, then the distance traveled every second is the same. The runner would cover a distance of 6 meters every second. If we could measure her position (distance from an arbitrary starting point) each second, then we would note that the position would be changing by 6 meters each second. This would be in stark contrast to an object that is changing its speed. An object with a changing speed would be moving a different distance each second. The data tables below depict objects with constant and changing speed.

Now let's consider the motion of that physics teacher again. The physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. The entire motion lasted for 24 seconds. Determine the average speed and the average velocity.

The physics teacher walked a distance of 12 meters in 24 seconds; thus, her average speed was 0.50 m/s. However, since her displacement is 0 meters, her average velocity is 0 m/s. Remember that the displacement refers to the change in position and the velocity is based upon this position change. In this case of the teacher's motion, there is a position change of 0 meters and thus an average velocity of 0 m/s.

Here is another example similar to what was seen before in the discussion of distance and displacement. The diagram below shows the position of a cross-country skier at various times. At each of the indicated times, the skier turns around and reverses the direction of travel. In other words, the skier moves from A to B to C to D.

Use the diagram to determine the average speed and the average velocity of the skier during these three minutes. When finished, click the button to view the answer.

As a last example, consider a football coach pacing back and forth along the sidelines. The diagram below shows several of coach's positions at various times. At each marked position, the coach makes a "U-turn" and moves in the opposite direction. In other words, the coach moves from position A to B to C to D.

What is the coach's average speed and average velocity? When finished, click the button to view the answer.

In conclusion, speed and velocity are kinematic quantities that have distinctly different definitions. Speed, being a scalar quantity, is the rate at which an object covers distance. The average speed is the distance (a scalar quantity) per time ratio. Speed is ignorant of direction. On the other hand, velocity is a vector quantity; it is direction-aware. Velocity is the rate at which the position changes. The average velocity is the displacement or position change (a vector quantity) per time ratio.

Visit Name That Motion.

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what is the difference between speed and a velocity

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Speed, being a scalar quantity, is the rate at which an object covers distance. The average speed is the distance (a scalar quantity) per time ratio. Speed is ignorant of direction. On the other hand, velocity is a vector quantity; it is a direction-aware quantity.

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Scegli per ciascuna risposta l'alternativa corretta. Quando hai completato l'esercizio, clicca su Conferma.

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Acceleration is a vector quantity that is defined as the rate at which an object changes its velocity. An object is accelerating if it is changing its velocity.

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Velocity is a vector quantity that refers to "the rate at which an object changes its position."
As such, velocity is direction aware.
When evaluating the velocity of an object, one must keep track of direction.

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Speed is a scalar quantity that refers to "how fast an object is moving." Speed can be thought of as the rate at which an object covers distance. A fast-moving object has a high speed and covers a relatively large distance in a short amount of time. Contrast this to a slow-moving object that has a low speed; it covers a relatively small amount of distance in the same amount of time. An object with no movement at all has a zero speed.

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Scalars are quantities that are fully described by a magnitude (or numerical value) alone.

Vectors are quantities that are fully described by both a magnitude and a direction.