Question

A bumper car with mass m₁ = 116 kg is moving to the right with a velocity of v₁ = 4 m/s. A second bumper car with mass m₂ = 97 kg is moving to the left with a velocity of v₂ = -3 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?

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

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

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

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

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?

Answer 1

m₁ = mass of first bumper car = 116 kg

V₁ = velocity of first bumper car towards right = 4 m/s

m₂ = mass of second bumper car = 97 kg

V₂ = velocity of second bumper car towards left = -3 m/s

1) velocity of center of mass is given as ::

V_cm = ( m₁ V₁ + m₂ V₂ ) / (m₁ + m₂)

V_cm = ( 116 x 4 + (97) (-3)) / (116 + 97)

V_cm = 0.8122 m/s         going right.

2) initial velocity of car 1 in ground reference = V1 = 4 m/s

velocity of car 1 in center-of-mass reference = V₁ - V_cm = 4 - 0.8122 = 3.188 m/s

3) Using conservation of momentum ::

m₁ V₁ + m₂ V₂ = m₁ V_1f + m₂ V_2f

116 x 4 + 97 (-3) = 116 V_1f + 97 V_2f

116 V_1f + 97 V_2f = 173        Eq-1

for elastic collsion ::

V_2f - V_1f = V₁ - V₂

V_2f = V_1f + 4 - (-3)

V_2f = V_1f + 7                     Eq-2

from Eq-1 and Eq-2

116 V_1f + 97 (V_1f + 7 ) = 173

V_1f = -2.4 m/s

in center of mass reference ::

final velocity of car 1 in center-of-mass reference = V_1f - V_cm = - 2.4 - 0.8122 = - 3.21 m/s

4. final velocity of car 1 in the ground (original) reference frame = V_1f = -2.4 m/s

5. Using Eq-2 V_2f = V_1f + 7  

V_2f = -2.4 + 7 = 4.6 m/s

6. Using conservation of momentum for inelastic collision ::

m₁ V₁ + m₂ V₂ = ( m₁ + m₂ )V

116 x 4 + 97 (-3) = (116 + 97 ) V

V = 0.8122 m/s