Question

segment 1 summative exam: forces & motion - version b
you are at an amusement park and are sitting at rest (v₁ᵢ) in a bumper car with a mass of 150 kg (m₁). your friend tim is in a 250 kg bumper car (m₂) and collides with your car at a speed of 5 m/s (v₂ᵢ). after the collision, your velocity is 6 m/s (v₁բ). the velocity of tims car after the collision is 1.4 m/s (v₂բ).

1. solve for the initial and final momentum of the system.

initial momentum
pᵢ = m₁v₁ᵢ + m₂v₂ᵢ
(150)(0)+(250)(5)
0 + 1250
final momentum
pբ = m₁v₁բ + m₂v₂բ
(150)(6)+(250)(1.4)
900 + 350
1250

1. explain how your answers support the law of conservation of momentum. (lt 1.2)

a. how did the momentum of each bumper - car change during the collision?
bumper car 1: the momentum didnt change.
1250 p
bumper car 2: the momentum didnt change.
1250 p
b. how does the change in velocity of each bumper car affect the total momentum of the system? explain using your data. the change in velocity of each bumper car affects the total momentum of the system by making it more or less.
c. use evidence to explain how this collision represents the law of conservation of momentum. is this an open or closed system? this collision represents the law of conservation of momentum because the initial momentum is equal to the final momentum after a collision in a closed system.

Explanation:

Step1: Calculate initial momentum

The initial velocity of your car $v_{1i}=0$ m/s, mass $m_1 = 150$ kg, and the initial velocity of Tim's car $v_{2i}=5$ m/s, mass $m_2 = 250$ kg. Using the formula for total initial momentum $p_i=m_1v_{1i}+m_2v_{2i}$, we have $p_i=(150\times0)+(250\times5)=0 + 1250=1250$ kg·m/s.

Step2: Calculate final momentum

Your final velocity $v_{1f}=6$ m/s and Tim's final velocity $v_{2f}=1.4$ m/s. Using the formula for total final momentum $p_f=m_1v_{1f}+m_2v_{2f}$, we get $p_f=(150\times6)+(250\times1.4)=900 + 350=1250$ kg·m/s.

Step3: Analyze momentum change of each car (for part 2A)

For bumper - car 1 (your car): Initial momentum $p_{1i}=m_1v_{1i}=150\times0 = 0$ kg·m/s, final momentum $p_{1f}=m_1v_{1f}=150\times6 = 900$ kg·m/s. The change in momentum $\Delta p_1=p_{1f}-p_{1i}=900 - 0=900$ kg·m/s. For bumper - car 2 (Tim's car): Initial momentum $p_{2i}=m_2v_{2i}=250\times5 = 1250$ kg·m/s, final momentum $p_{2f}=m_2v_{2f}=250\times1.4 = 350$ kg·m/s. The change in momentum $\Delta p_2=p_{2f}-p_{2i}=350 - 1250=- 900$ kg·m/s.

Step4: Analyze velocity - momentum relationship (for part 2B)

The change in velocity of your car $\Delta v_1=v_{1f}-v_{1i}=6 - 0 = 6$ m/s and for Tim's car $\Delta v_2=v_{2f}-v_{2i}=1.4 - 5=-3.6$ m/s. According to the law of conservation of momentum $m_1\Delta v_1=-m_2\Delta v_2$. The increase in your car's momentum due to the increase in velocity is compensated by the decrease in Tim's car's momentum due to the decrease in velocity, keeping the total momentum of the system constant.

Step5: Determine system type (for part 2C)

This is a closed - system collision because there is no external force acting on the two - car system during the short time of the collision. The initial momentum $p_i = 1250$ kg·m/s is equal to the final momentum $p_f = 1250$ kg·m/s, which is in accordance with the law of conservation of momentum.

Answer:

1. Initial momentum of the system: 1250 kg·m/s, Final momentum of the system: 1250 kg·m/s

2.
A. Bumper car 1: Initial momentum $p_{1i}=0$ kg·m/s, final momentum $p_{1f}=900$ kg·m/s, change in momentum $\Delta p_1 = 900$ kg·m/s. Bumper car 2: Initial momentum $p_{2i}=1250$ kg·m/s, final momentum $p_{2f}=350$ kg·m/s, change in momentum $\Delta p_2=-900$ kg·m/s.
B. The change in velocity of each car results in a change in their individual momenta, but the sum of the momenta of the two cars remains constant because the increase in one car's momentum is equal to the decrease in the other car's momentum.
C. This is a closed - system collision as there is no external force during the collision and the initial momentum (1250 kg·m/s) is equal to the final momentum (1250 kg·m/s), which follows the law of conservation of momentum.