Collisions

Convergence, Jackson Pollock, 1952

AP Topic 4.4

Describe whether an interaction between objects is elastic or inelastic.

Collision

• An interaction between objects where momentum and energy are transferred.

Elastic Collisions

• A collision in which both momentum AND kinetic energy are conserved.
• No collision is actually perfectly elastic. However, to simplify some collisions, we can consider them to be perfectly elastic (“close enough” such as force carts, billiard balls, etc.)

Elastic Collisions

p_i = p_f

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

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K_i = K_f

½m₁v_1i² + ½m₂v_2i² = ½m₁v_1f² + ½m₂v_2f²

Inelastic Collisions

In ALL collisions, momentum is conserved.

However, in an inelastic collision kinetic energy is not conserved. There is a loss of kinetic energy.

Kinetic energy is transformed (e.g., thermal energy, sound energy, and material deformation.)

Inelastic Collisions

p_i = p_f

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

​

K_i > K_f

Inelastic Collisions

Two cars colliding would be an inelastic collision, even if they bounce off each other. Why?

If two objects collide, stick, and move as one object, the collision would be considered perfectly inelastic (see below). This results in the maximum loss of kinetic energy.

Example 1

Cart 1 with a mass of 30 g is moving to the right at 12 m/s on a track when it collides with cart 2 which is stationary and has a mass of 24 g. After the collision, cart 1 has a velocity of 1.33 m/s to the right and cart 2 has a velocity of 13.33 m/s to the right. Is this an elastic or inelastic collision?

30 g

24 g

12 m/s

Before Collision

After Collision

30 g

24 g

1.33 m/s

13.33 m/s

Example 2

Object X and object Y each have a mass of 2 kg and collide at 0.3 s as shown in the graph above. Comment on the conservation of momentum, the conservation of kinetic energy, and the type of collision between the two objects.

Example 3

A 2 kg object traveling at 5 m/s on a frictionless horizontal surface collides head-on with and sticks to a 3 kg object initially at rest. 

1. What type of collision is this?
2. What happens to the kinetic energy of the 2-object system after the collision?
3. What is the speed of the 2-object system after the collision?

Example 3

A 2 kg object traveling at 5 m/s on a frictionless horizontal surface collides head-on with and sticks to a 3 kg object initially at rest. 

1. What type of collision is this? inelastic
2. What happens to the kinetic energy of the 2-object system after the collision?
3. What is the speed of the 2-object system after the collision?

Example 3

A 2 kg object traveling at 5 m/s on a frictionless horizontal surface collides head-on with and sticks to a 3 kg object initially at rest. 

1. What type of collision is this? inelastic
2. What happens to the kinetic energy of the 2-object system after the collision? it decreases
3. What is the speed of the 2-object system after the collision?

Example 3

A 2 kg object traveling at 5 m/s on a frictionless horizontal surface collides head-on with and sticks to a 3 kg object initially at rest. 

1. What type of collision is this? inelastic
2. What happens to the kinetic energy of the 2-object system after the collision? it decreases
3. What is the speed of the 2-object system after the collision? 2 m/s

Practice

In a circus act, a cannon ball (b) of mass m_b is fired with a velocity of v_b at the back of a stationary clown car (c) of mass 20m_b. The cannon ball then lodges itself into the car. After the collision, the ball-car system moves forward. Friction cannot be ignored.

a) On the axes to the right, sketch a graph of the total kinetic energy, K_b-c, of the ball-car system from the moment the ball is launched through the system’s complete motion. The collision occurs at t_c and continues until t_max.

Practice

A cannon ball (b) of mass m_b is fired with a velocity of v_b at the back of a stationary clown car (c) of mass 20m_b. The cannon ball then lodges itself into the car. After the collision, the ball-car system moves forward. Friction cannot be ignored.

b) Derive an expression for the speed of the ball-car system just after the collision of the ball with the car.

Practice

Three more acts are performed, one at a time, with the same mass of different types of cannonballs fired at the same speed to the same type of clown car.

• In the first act, the ball sticks to the car and the car moves a distance d₁.
• In the second act, the ball bounces off the car perfectly elastically. The ball bounces backward and the car moves forward a distance d₂.
• In the third act, the ball goes completely through the car (no clowns were injured) and continues moving forward. The car moves forward a distance d₃.

c. Sketch a graph of the total kinetic energy, K_b-c, of the ball-car system from the moment the ball is launched through the system’s complete motion for Acts 2 and 3 like you did for Act 1.

Practice

Three more acts are performed, one at a time, with the same mass of different types of cannonballs fired at the same speed to the same type of clown car.

• In the first act, the ball sticks to the car and the car moves a distance d₁.
• In the second act, the ball bounces off the car perfectly elastically. The ball bounces backward and the car moves forward a distance d₂.
• In the third act, the ball goes completely through the car (no clowns were injured) and continues moving forward. The car moves forward a distance d₃.

d. Indicate in which of the three acts the collision resulted in the maximum distance travelled by the car. Justify your answer.