B.2.1.2 Momentum changes and collisions in sports scenarios (HL) Notes

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)

1. In any collision:
   1. Momentum changes due to applied forces
   2. The change in momentum equals the impulse
   3. Total momentum is conserved in a closed system

Principles of Momentum in Collisions:

1. Every collision involves:
   1. Change in momentum for both objects
   2. Transfer of energy between objects
   3. Varying degrees of elasticity
2. The coefficient of restitution (CR) determines:
   1. How much objects "bounce" after collision
   2. Energy retained vs energy lost
   3. Effectiveness of sports equipment
3. Conservation of Momentum: In a closed system, total momentum before a collision = total momentum after a collision.
4. Impulse (Force × Time): A longer force application increases momentum change (e.g., following through in a golf swing).

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 hitting court
3. Perfectly Inelastic Collisions:
   1. Objects stick together
   2. Maximum energy loss
   3. Example: Catching a ball

Applications in Sport

1. Equipment Design:
   1. Tennis racquets with optimal string tension
   2. Golf club faces with regulated CR
   3. Basketball with standardised bounce height
2. Performance Analysis:
   1. Ball speed after impact
   2. Energy transfer in collisions
   3. Equipment efficiency

Principles of angular movement

1. A force applied to a rotating object that also does not act through the axis (center of rotation) creates torque
2. This force is called an eccentric force.
3. The size of the torque created depends on the size of the force, the direction of the force, and how far the force is applied from the axis of rotation.

Coefficient of Restitution

Factors Affecting COR

1. Material Properties

1. Harder materials (e.g., metal, rubber) have a higher COR.
2. Softer materials (e.g., leather, foam) absorb more energy, leading to 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.

Applications of COR in Sports

1. Basketball: The air pressure in a ball affects its COR, determining bounce height and control.
2. Tennis: Tennis balls lose COR over time, affecting their rebound on different court surfaces.
3. Golf: The COR of a golf ball and clubface affects shot distance and ball speed.
4. Baseball & Softball: Aluminum bats have a higher COR than wooden bats, increasing ball exit velocity.
5. Soccer: COR influences how the ball rebounds off the goalpost or a player's foot.

Calculating the Coefficient of Restitution

1. The formula for CR is:
2. CR = Velocity After Impact / Velocity Before Impact
3. Values range from 0 (perfectly inelastic, no bounce) to 1 (perfectly elastic, full bounce-back).

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

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.