Question

A railroad car of mass M moving at a speed v1 collides and couples with two coupled railroad cars, each of the same mass M and moving in the same direction at a speed v2.

(a) What is the speed vf of the three coupled cars after the collision in terms of v1 and v2? vf =

(b) How much kinetic energy is lost in the collision? Answer in terms of M, v1, and v2. ?KE =

Answer 1

To find the final velocity vf of three coupled cars after an inelastic collision, use the conservation of momentum equation. The loss in kinetic energy ΔKE is calculated by subtracting the final kinetic energy from the initial kinetic energy. These calculations involve the initial velocities and mass of the railroad cars.

Step-by-step explanation:

The scenario involves an inelastic collision where a single railroad car collides and couples with two other railroad cars, and all cars have the same mass M. To find the final velocity (vf) of the three coupled cars after the collision, we can use the conservation of momentum, which in formula form is:

Mv1 + 2Mv2 = 3Mvf

Solving for vf gives:

vf = (Mv1 + 2Mv2) / 3M

vf = (v1 + 2v2) / 3

To calculate the loss in kinetic energy (ΔKE), we need to consider the kinetic energies before and after the collision. Initially, the kinetic energy is:

KEi = (1/2)Mv1^2 + (1/2)(2M)v2^2

KEi = (1/2)Mv1^2 + Mv2^2

After the collision, the kinetic energy of the combined mass (3M) moving with velocity vf is:

KEf = (1/2)(3M)vf^2

KEf = (1/2)(3M)((v1 + 2v2) / 3)^2

Subtracting the final kinetic energy from the initial kinetic energy gives the kinetic energy lost:

ΔKE = KEi - KEf

ΔKE = [(1/2)Mv1^2 + Mv2^2] - (1/2)(3M)((v1 + 2v2) / 3)^2

Answer 2

Answer:
a.
`v_f=(Mv_1+2Mv_2)/(3M)`

b.
`KE_i-KE_f=(M(v_1-v_2)^2)/(3)`

Step-by-step explanation:

These are the correct answers (No explanation).