Question

An automobile traveling at 60 km/h bumps into the rear of another automobile traveling at 55 km/h in the same direction. The mass of the first automobile is 1200 kg, and the mass of the second automobile is 1000 kg. If the collision is elastic, find the velocities of both automobiles immediately after this collision. (Hint : Solve this problem in a reference frame moving with a velocity equal to the initial velocity of one of the automobiles.)

Please show all work, I am especially confused about the reference frame so please explain this in detail. Thanks

Answer 1

Initial velocities are:

uA = 60 km/h = 16.667 m/s

uB = 55 km/h = 15.277 m/s

In A's reference frame, the speed of B is: uB' = 15.277 - 16.667 = - 1.389 m/s

Also, in A's reference frame, its own speed is 0 m/s

so, using conservation of momentum

0 - 1000(1.389) = 1200v1 + 1000v2

=> 12v1 + 10v2 = 13.89

=> v2 = 1.389 - 1.2v1 ..........................[1]

Also, since the collision is elastic, the kinetic energy is conserved as well.

So,

=> .....................[2]

substitute [1] in [2] to get:

=>

solving this quadratic equation gives, v₁ = 1.263 m/s and v'₁ = - 0.00024 m/s

so, final velocity of A (in rest frame) is 16.667 - 0.00024 = 16.66676 m/s and final velocity of B (in rest frame) is 17.93 m/s.