Question

A bumper car with mass m1 = 106.0 kg is moving to the right with a velocity of v1 = 4.5 m/s. A second bumper car with mass m2 = 98.0 kg is moving to the left with a velocity of v2 = 3.1 m/s. The two cars have a completely elsatic collision. Assume the surface is frictionless.

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

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

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

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

Answer 1

Since we have a elastic collision,

solving (1) for

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

or

solving (1) for

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

or

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

According to Galilean transformation,

isolating the final speed of car 1 in the center of mass reference frame

where is the final velocity of the center of mass and is given by the equation,

or

inserting in eq. (2),

or

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

Use Galilean transformation again,

isolating the initial speed of car 1 in the center of mass reference frame

where is the initial velocity of the center of mass and is given by the equation,

or

Notice that the intial velocity of the center of mass reference frame remains the same. This was expected because there are not external forces acting on the center of mass. So,

or