IB Sports, exercise and health science B.2.1.3 Momentum Changes and Collisions in Sports Scenarios Notes

Collisions in Sport

1. A collision happens when two or more bodies meet and push on each other for a brief moment, often only a few thousandths of a second, and is enough to change the motion of both bodies.
2. Sport is full of collisions: a foot striking a football, a bat meeting a ball, a hand pulling against water, a sprinter's spikes pressing into the track.

Momentum and Impulse: The Building Blocks of Collisions

1. Momentum is the quantity of motion an object has.
2. It depends on two factors: mass and velocity: $p=mv$

Where:

• p = momentum (kg·m/s)
• m = mass (kg)
• v = velocity (m/s)

Principles of Momentum in Collisions:

1. Conservation of Momentum: In a closed system, total momentum before a collision = total momentum after a collision.
2. The link to Newton's third law: the two bodies push on each other with forces equal in size and opposite in direction for the same time, so the impulses are equal and opposite and the changes in momentum cancel out, leaving total momentum unchanged.
3. Impulse (Force × Time): A longer force application increases momentum change (e.g., following through in a golf swing).
4. One-dimensional calculations: pick a positive direction first, write each velocity with the right sign, then apply m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ where u are velocities before and v are velocities after.

Types of Collisions in Sport

1. Elastic Collisions:
   1. Maximum energy retention
   2. High coefficient of restitution
   3. Example: Pool balls colliding
2. Inelastic Collisions:
   1. Energy partially converted to heat/sound
   2. Lower coefficient of restitution
   3. Example: Tennis ball bouncing on court (CR around 0.5 to 0.75)
3. Perfectly Inelastic Collisions:
   1. Objects stick together
   2. Maximum energy loss
   3. Example: Catching a ball

Coefficient of Restitution

Calculating the Coefficient of Restitution

1. The formula for CR is:
2. CR = (speed of separation) / (speed of approach)
3. Values range from 0 (perfectly inelastic, no bounce) to 1 (perfectly elastic, full bounce-back).

$$e = \frac{v_2 - v_1}{u_1 - u_2}$$

1. The numerator (v_b − v_a) is the speed of separation after the collision.
2. The denominator (u_a − u_b) is the speed of approach before the collision.
3. CR = 1 is a perfectly elastic collision with no kinetic energy lost, but no real sporting collision actually reaches 1.
4. CR = 0 is a perfectly inelastic collision where the bodies stick together, such as a clay ball thrown at a wall.

Factors Affecting COR

1. Material Properties

1. Materials that retain elastic energy on contact, such as rubber or a tightly-strung composite, have a higher COR.
2. Materials that absorb energy as heat or deformation, such as soft foam or wet clay, have a lower COR.

2. Surface Conditions

1. A hard surface (e.g., concrete) increases COR compared to a soft surface (e.g., grass).
2. Wet or rough surfaces can decrease COR due to increased energy absorption.

3. Temperature

1. Warmer objects (e.g., heated tennis balls) have a higher COR because their materials become more elastic.
2. Cold temperatures reduce elasticity, decreasing COR (e.g., a frozen baseball bounces less).

4. Impact Velocity

1. Higher impact speeds can deform objects, reducing COR due to energy loss as heat and sound.

Practical Applications of Momentum, Impulse, and CR

1. Sports Equipment Design

1. Golf Clubs: High CR clubfaces (e.g., trampoline designs) increase ball speed.
2. Tennis Balls: CR is regulated to ensure consistent bounce and gameplay.

2. Performance Optimization

1. High Jump: Athletes maximize impulse by applying force over a longer time during take-off.
2. Boxing: Gloves increase the time of impact, reducing the force felt by the opponent.

3. Safety Engineering

1. Car Crumple Zones: Increase impact time to reduce force and protect passengers.
2. Helmets: Absorb energy, reducing the momentum transferred to the head.