A bumper car with mass m 1 = 100 kg is moving to the right with a velocity of v

Question

A bumper car with mass m1 = 100 kg is moving to the right with a velocity of v1 = 4.5 m/s. A second bumper car with mass m2 = 85 kg is moving to the left with a velocity of v2 = -3.7 m/s. The two cars have an elastic collision. Assume the surface is frictionless.

What is the velocity of the center of mass of the system?

Answer .7324

What is the initial velocity of car 1 in the center-of-mass reference frame?

3.7676

What is the final velocity of car 1 in the center-of-mass reference frame?

-3.7676

What is the final velocity of car 1 in the ground (original) reference frame?

need help with this

What is the final velocity of car 2 in the ground (original) reference frame?

need help with this

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?

need help with this

Explanation / Answer

a) Speed of Center of mass = (m1v1 + m2v2)/(m1 + m2) = ((100 * 4.5) + (85 * -3.7))/(100+85) = 0.732 m/s

b) Initial velocity of car 1 in center of mass frame = (4.5 - 0.732) m/s = 3.768 m/s

When they have an elastic collision let final speed of car 1 be v1 and that of car 2 be v2

hence

(100 * 4.5) - (85 * 3.7) = 100v1 + 85v2

100v1 + 85v2 = 135.5 ---(1)

coefficient of restitution equation

1 = -(v1 - v2)/(4.5 + 3.7)

v2 - v1 = 8.2 ----(2)

Solving (1) and (2) we get:

v1 = -3.036 m/s (negative means opposite to the direction we have assumed)

v2 = 5.164 m/s

Center of mass will continue to move in the same direction with same speed even after the collision because momentum is conserved

hence

c) Final velocity o fcar 1 in center of mass frame = -3.036 - 0.732 = -3.768 m/s (-ve means opposite to the direction in it was traveling)

d) Final velocity of car1 in ground frame = v1 = -3.036 m/s

e) Final velocity of car2 in ground frame = v2 = 5.164 m/s

f) The coefficient of restitution equation that I have written put e = 0 for inelastic equation you will get a different equation (2). Solve (1) and (2) to get the results :)

Please rate for explanation :)

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