A bumper car with mass m1 = 100 kg is moving to the right with a velocity of v1 = 4.6 m/s. A second bumper car with mass m2 = 86 kg is moving to the left with a velocity of v2 = -3.8 m/s. The two cars have an elastic collision. Assume the surface is frictionless.
1) What is the velocity of the center of mass of the system?
m/s
2) What is the initial velocity of car 1 in the center-of-mass reference frame?
m/s
3) What is the final velocity of car 1 in the center-of-mass reference frame?
m/s
4) What is the final velocity of car 1 in the ground (original) reference frame?
m/s
5) What is the final velocity of car 2 in the ground (original) reference frame?
m/s
6) In a new (inelastic) collision, the same two bumper cars with the same initial velocities now latch together as they collide.
What is the final speed of the two bumper cars after the collision?
1) 0
To solve this problem, we can use the principle of conservation of momentum.
1) The velocity of the center of mass of the system can be calculated using the formula for the center of mass:
Vcm = (m1 * v1 + m2 * v2) / (m1 + m2)
= (100 kg * 4.6 m/s + 86 kg * (-3.8 m/s)) / (100 kg + 86 kg)
2) The initial velocity of car 1 in the center-of-mass reference frame is equal to the relative velocity of car 1 with respect to the center of mass. In this case, both cars are moving in opposite directions, so:
V1_initial_relative = v1 - Vcm
3) The final velocity of car 1 in the center-of-mass reference frame can be calculated using the conservation of momentum. Since the collision is elastic, the total momentum before the collision is equal to the total momentum after the collision:
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
4) To calculate the final velocity of car 1 in the ground reference frame, we need to consider the change in velocity due to the velocity of the center of mass:
V1_final = V1_final_reference_frame + Vcm
5) The final velocity of car 2 in the ground reference frame can be calculated in a similar way:
V2_final = V2_final_reference_frame + Vcm
6) For the inelastic collision, the final speed of the latched together cars can be calculated using the conservation of momentum:
(m1 + m2) * v_final = m1 * v1 + m2 * v2
Let's calculate the values step-by-step:
1) The velocity of the center of mass is calculated as:
Vcm = (100 kg * 4.6 m/s + 86 kg * (-3.8 m/s)) / (100 kg + 86 kg)
= (460 kg*m/s - 324.8 kg*m/s) / 186 kg
= 135.2 kg*m/s / 186 kg
≈ 0.726 m/s
Therefore, the velocity of the center of mass of the system is approximately 0.726 m/s.
2) The initial velocity of car 1 in the center-of-mass reference frame is:
V1_initial_relative = v1 - Vcm
= 4.6 m/s - 0.726 m/s
≈ 3.874 m/s
Therefore, the initial velocity of car 1 in the center-of-mass reference frame is approximately 3.874 m/s.
3) To calculate the final velocity of car 1 in the center-of-mass reference frame, we use the conservation of momentum:
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
Rearranging the equation to solve for v1_final:
v1_final = (m1 * v1_initial + m2 * v2_initial - m2 * v2_final) / m1
Substituting the given values:
v1_final = (100 kg * 4.6 m/s + 86 kg * (-3.8 m/s) - 86 kg * v2_final) / 100 kg
Multiplying through by 100 kg:
100 kg * v1_final = 460 kg*m/s - 324.8 kg*m/s - 86 kg * v2_final
100 v1_final = 1352 - 324.8 - 86 v2_final
100 v1_final + 86 v2_final = 1027.2
Simplifying:
10 v1_final + 8.6 v2_final = 102.72
4) To calculate the final velocity of car 1 in the ground reference frame, we use the equation:
V1_final = V1_final_reference_frame + Vcm
Substituting the calculated value for Vcm and the previously calculated v1_final:
V1_final = v1_final + Vcm
= v1_final + 0.726 m/s
5) To calculate the final velocity of car 2 in the ground reference frame, we use the equation:
V2_final = V2_final_reference_frame + Vcm
Substituting the calculated value for Vcm and the previously calculated v2_final:
V2_final = v2_final + Vcm
6) For the inelastic collision, the final speed of the latched together cars can be calculated using the conservation of momentum:
(m1 + m2) * v_final = m1 * v1 + m2 * v2
Rearranging the equation to solve for v_final:
v_final = (m1 * v1 + m2 * v2) / (m1 + m2)
Substituting the given values:
v_final = (100 kg * 4.6 m/s + 86 kg * (-3.8 m/s)) / (100 kg + 86 kg)
Calculating:
v_final = (460 kg*m/s - 324.8 kg*m/s) / 186 kg
= 135.2 kg*m/s / 186 kg
≈ 0.726 m/s
Therefore, the final speed of the two bumper cars after the inelastic collision is approximately 0.726 m/s.
To solve these questions, we can use the conservation laws of momentum and kinetic energy for the given scenarios.
1) The velocity of the center of mass of the system can be calculated using the formula:
V_cm = (m1 * v1 + m2 * v2) / (m1 + m2)
Substitute the given values:
V_cm = (100 kg * 4.6 m/s + 86 kg * (-3.8 m/s)) / (100 kg + 86 kg)
2) The initial velocity of car 1 in the center-of-mass reference frame is zero because the velocity of the center of mass is extensively chosen as the reference frame.
3) For an elastic collision, the relative velocity of the two cars before and after the collision in the center-of-mass reference frame remains the same. So, the final velocity of car 1 in the center-of-mass reference frame is equal to the negative of its initial velocity, which means -v1.
4) The final velocity of car 1 in the ground (original) reference frame can be calculated by adding the velocity of the center of mass to the final velocity of car 1 in the center-of-mass frame.
V_final_car1 = V_cm + final_velocity_car1_in_center_of_mass_frame
5) The final velocity of car 2 in the ground (original) reference frame can also be calculated similarly, by adding the velocity of the center of mass to the final velocity of car 2 in the center-of-mass frame.
V_final_car2 = V_cm + final_velocity_car2_in_center_of_mass_frame
6) In an inelastic collision, the two cars stick together after the collision. To find the final speed, we need to conserve momentum and kinetic energy.
Since the two cars latch together, the total mass after the collision would be the sum of their masses: m1 + m2.
The momentum before the collision is given by P = m1 * v1 + m2 * v2.
The final speed can be calculated using the equation:
final_speed = P / (m1 + m2)
To find the specific values, you can substitute the given numbers into the formulas and perform the necessary calculations.