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Collisions occur in sport all the time. Whether it is a ball hitting a bat, a ball bouncing on the ground or two players bumping into each other, the outcome of the collisions often determine the result of many sporting contests. Fortunately, using a couple of relatively simple physics concepts, it is often possible to predict what will happen when two objects collide

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The coefficient of restitution is the ratio of the speed of a ball directly after a bounce to its speed before a bounce. As COR is a ratio, it has no units.
e = v ₂/v ₁

where e = the coefficient of restitution
v2 = the speed after the bounce
v1 = the speed before the bounce

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The coefficient of restitution actually depends on the ball and the surface it is bouncing on. A tennis ball bouncing on grass has a different COR than a tennis ball on bouncing on clay. If there is no energy lost in the bounce, the ball would move just as fast after the bounce as it did before it, and e would be equal to 1.

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A coefficient of greater than 1 would mean that the ball had more energy after the bounce than before it, which violates the law of conservation of energy.

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Since the COR is defined in terms of speed, v, and kinetic energy, Ek, is proportional to velocity squared, then :
e = v ₂/v ₁ = √(Ek)₂/(Ek)₁

Consider a ball dropped to the ground from a height H and rebounding to a height h. According to the law of conservation of energy, the kinetic energy of the ball as it hits the ground is the same as the gravitational potential energy of the ball at the top of the bounce. Therefore:
e = v ₂/v ₁ = √(Ek)₂/(Ek)₁ = √(Eg)₂/(Eg)₁ = √mgh/mgH = √h/H

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So the coefficient of restitution can be calculated from the initial drop height and the height of the first bounce. This is how many sports define the coefficients of restitution of the balls used to play them.

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When an object slides over a rough surface, friction opposes its motion. The size of the frictional force depends on the nature of the surfaces and the size of the normal force. The relationship between the force of friction, F f, and the normal force, F N, is as expressed as:
F f = μF N
The term μ is the coefficient of friction. Its value depends on the type of surface involved and whether it is rough, polished, wet, dry, gravelly, etc. A low coefficient of friction indicates a small degree of friction

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Frictional forces can be measured in different situations, leading to kinetic friction and static friction
Kinetic friction (a.k.a sliding friction): applies when one object is moving across a surface. The coefficient of kinetic friction is represented with the symbol μ k
Static Friction: the force that keeps an object stationary even while a pushing or pulling force acts on it. The coefficient of static friction is represented with the symbol μ st
Note*: Static friction coefficients will always be larger than kinetic coefficients. This is because the forces that act between the surfaces are weakened when the surfaces are moving past each other, and the bonds are not able to form as strongly.

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Differences between kinetic and static friction

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Differences between kinetic and static friction

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When a ball slides, it experiences kinetic friction because the same part of the ball touches the ground all the time
When that ball starts rolling, it then experiences static friction since only one section of the ball touches the ground at a time

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When looking at the motion of a ball, it is often easier to treat it as a point mass (a particle with mass but no other dimensions). For example, when considering the motion of a tennis ball, it is easier to assume that the entire mass of the ball is concentrated on a single point at the centre of the ball. This simplifies the situation immensely, since you have to only consider the ball's translational motion (the object's motion from one point in space to another). In reality though, tennis players are going to try to hit the ball so that it spins in a way that makes it more difficult for their opponent to hit. In order to analyse the spinning motion of the the ball, you have to consider its rotational motion

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Rotational motion is measured in angles, as you need to use point mass to measure the objects rotation from its origin to the next point. The angles themselves aren't measured in degrees, as we have come to expect. The angles are measured in radians, as physicists prefer to use this kind of measurement.
1 revolution (360⁰) = 2π radians
So 180⁰ = π radians
Which = 180⁰/π = 57.3⁰

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Currently, the ball has only been measured in linear speed (movement from one point to the another). To measure the speed of its rotation, you must figure out its angular speed.
Angular speed is defined as:
w = θ/t
where w is angular speed
θ is the angle
t is the time
If time is measured in seconds (s) and angle is measured in radians (rad), then the units for angular speed are rad s⁻¹
If the ball is spinning too rapidly, it can be easier to measure the frequency of the ball's rotation. Frequency (f) is measured in Hertz (Hz) where 1 Hz = 1 rotation per second.
Frequency can be easily converted into a time of rotation using this formula:
w = 2π/t = 2πf
where f is the frequency (in Hz or rotations per second

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Converting between angular speed and linear speed

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Converting between angular speed and linear speed

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The linear speed of a rotating ball can be calculated from its angular speed if the radius of the ball is known. The formula to convert angular speed to linear speed is:
v = rw
where v is linear speed (in m s⁻¹)
r is the radius of the ball (in m)
w is the angular speed (in rad s⁻¹)
If the angular speed w, was measured in degrees per second rather than radians per second, then the conversion formula would be:
v = w/360 x 2πr

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A simple pendulum is made up of a mass (bob) attached to a piece of string, wire or rope at a fixed point. When this bob is moved slightly to one side and released, it will swing back and forth unless disturbed. This type of motion is known as simple harmonic motion. As the pendulum is in motion, it is an example of the conversion of mechanical energy

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The Magnus Effect explains why a ball will react differently with air when it is spinning, in comparison to when it is dropped straight down
Here is a video explaining how the air pushes a spinning ball forward, rather than falling flat
https://youtu.be/2OSrvzNW9FE

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The Bernoulli Principle (How planes fly, supposedly)

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The Bernoulli Principle (How planes fly, supposedly)

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Bernoulli is a scientist who had a theory to rival that of Newton's law of equal and opposite reactions. Newton's theory states that airplanes fly because wings deflect air downward, so that in reaction the plane is forced upward. Bernoulli challenged this theory, by saying that airplanes fly because the pressure above the wing of the plane is smaller than the pressure below the wing
Here is a video explaining how Bernoulli came up with this theory, and how it relates to the Magnus Effect's change in movement
https://youtu.be/YKkYAPA04ZY

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Pearson Places http://www.pearsonplaces.com.au/Places/My_Passport.aspx
Aviation for kids (Magnus Effect) http://www.aviation-for-kids.com/the-magnus-force.html, (Bernouli Principle) http://www.aviation-for-kids.com/Lift.html
Magnus Effect (Detailed understanding - Human Kinematics
http://www.humankinetics.com/excerpts/excerpts/magnus-effect-

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Youtube: Veritasiuem (Magnus Effect) https://youtu.be/2OSrvzNW9FE
Hewitt-Drew-it! PHYSICS (Bernoulli Principle) https://youtu.be/YKkYAPA04ZY

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Background Diagram: Motion sensors in sport http://www.americanfootballinternational.com/motion-sensors-continue-technological-advances-sport/
Double Pendulum Diagram:
https://www.revolvy.com/main/index.phps=Double%20Pendulum&item_type=topic
Physics textbook: Flight of a ball and position-time graphs

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Different energy forms in a simple pendulum

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Different energy forms in a simple pendulum

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If you take the lowest point of a swinging bob's as a reference point and mark it as the zero potential energy, the bob then goes through multiple energy forms. At the top of the bob's swing, the bob will have its maximum potential energy (Eg). At the bottom of the bob's swing, all the gravitational energy is converted into kinetic energy (Ek). This is also known as the bob's maximum velocity. As the bob completes its swing, the kinetic energy is converted back into potential energy. It is crucial that the bob changes from potential to kinetic energy, as otherwise it wouldn't be in constant motion. If it were to use only kinetic energy, more and more of it would be lost due to heat and sound, resulting in the bob stopping.

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Bonus: How cricketers find the 'Hot Spot' on a camera

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Bonus: How cricketers find the 'Hot Spot' on a camera

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When a cricket ball hits a cricket bat, there's a small amount of friction between the ball and bat making contact. This friction results in the production of some heat, which is how thermal imaging sees the heat signature of the ball on the bat.

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The period of a pendulum is the time it takes for the bob to complete one full swing (oscillation)
The period of the pendulum is given by the formula:
T = 2π√L/g
where T is the period of oscillation (in s)
L is the length of the simple pendulum (in m)
g is the acceleration due to gravity (9.8m s⁻²)
When calculating the period of a pendulum, the only variable is the length of the pendulum

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Most objects involved with locomotion are much more complicated than a simple pendulum. A double pendulum consists of a bob attached to another bob, which is connected to a fixed point (a simple pendulum on top of another simple pendulum, virtually).
The limbs in your body are a good example of these double pendulums. Your lower arm (Ulna/Radius) is connected to your upper arm (Humerus) in a joint; if you swing your lower arm forward, your upper arm will be carried by it. Same with your legs, your lower leg (Fibula/Tibia) to your upper leg (Femur)
Below is a visual of what a double pendulum consists of. m ₁ being the first bob, m ₂ being the second, lower bob

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To simplify physics equations involving movement or motion, the effects of air resistance are usually ignored. But obviously, in reality a ball will move through the air, experiencing a retarding force due to fluid friction (as air is regarded as a fluid). This is known as air resistance, or drag. If someone were to throw a ball without the presence of drag, the ball would follow a perfect parabolic path due to gravity

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Investigating movement in two dimensions

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Investigating movement in two dimensions

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When analyzing the flight of an object, the movement needs to be measured in both horizontal and vertical planes. This graph shows a single bounce of a tennis ball in relation to the horizontal and vertical axis, and how its biggest change in velocity is right after the first bounce and right before the second bounce

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What can be gathered from a horizontal graph

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What can be gathered from a horizontal graph

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The things that can be gathered from this graph are:
The distance from the origin, as it will increase the further the ball travels
The graph is a straight line, indicating that the ball was constantly moving away from its origin

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What can be gathered from a vertical graph

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What can be gathered from a vertical graph

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What can be gathered from the vertical position-time graph is this:
The distance from the origin increases as the ball gets higher, and becomes lower as the ball approaches the ground
The vertical graph is parabolic, with the points becoming closer as the graph curves

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Applying equations of motion to the flight of a ball

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Applying equations of motion to the flight of a ball

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When applying motion equations to a two-dimensional plane, it is viewed as a more complex version of the normal, one-dimensional equations, except you have to factor for the horizontal and vertical components, as they are working together.
When calculating horizontal motion, no forces are being applied, and the object will have an acceleration of 0
So v = u + at becomes v = u
The same applies to the other equations of motion
s = ut = vt
When vertical motion is calculated, the acceleration is 9.8 m s⁻² by default, as the ball is rising and falling due to gravity.
v = u + 9.8t
v ² = u ² + 19.6s
s = 1/2(u + v)t
s = ut + 4.9t ²
To calculate the two-dimensional velocity, just add the vertical and horizontal components together

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It is difficult to calculate the path of a ball affected by air resistance as drag is not a constant force, and it has many variables
Mathematically, the drag force can be calculated by the following equation:
F D = 1/2C D pv ² A
where F D = the drag force (in N)
v = the velocity of the ball (in m s⁻¹)
p = the density of the air (in kg m⁻³)
A = the cross-sectional area of the ball (in m²)
C D is a dimensionless constant (a.k.a the drag coefficient) that varies depending on the shape of the ball. This is usually done by conducting experiments in a wind tunnel. For a normal, smooth ball the C D is approximately 0.5, but most balls in sport have stitching or ripples on the surface, reducing their overall drag. This is due to the balls catching air and creating a layer of air around it, making it easier for a ball to pass through the air. So while a smooth ball has a C D of 0.5, a baseball has a C D of around 0.35, golf balls having even less drag at 0.2.

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Every object, when dropped from a great height, will eventually accelerate to the point where the object begins to travel at a constant speed. This is also where the drag force is equal to the weight of the ball, and it is known as an object's terminal velocity
At terminal velocity, F D = F g
Therefore, for a ball:
1/2C D pv ² A = mg
v t ² = 2mg/C D pA
v t = √2mg/C D pA
v t is terminal velocity (in m s⁻¹)
m is the mass of the falling object (in kg)
g is the acceleration due to gravity (9.8 m s⁻² down)