Question

Cart 111 of mass mmm is traveling with speed v_0v 0 ​ v, start subscript, 0, end subscript in the + x+xplus, x-direction when it has an elastic collision with cart 222 of mass 3m3m3, m that is at rest. What are the velocities of the carts after the collision?

Answer 1

v1 = -v0/3 ,v2 = 2v0/3

Step-by-step explanation:

Answer 2

In an elastic collision, the momentum and the kinetic energies are conserved.

Momentum:

`\vec{P_i} = \vec{P_f}\\\vec{P}_1 + \vec{P}_2 = \vec{P}_1' + \vec{P}_2'\\m\vec{v_0} + 0 = m\vec{v_1}' + 3m\vec{v_2}}' \\v_0 = v_1 + 3v_2`

Kinetic energy:

`K_i = K_f\\K_1 + K_2 = K_1' + K_2'\\(1)/(2)mv_0^2 + 0 = (1)/(2)m{v_1'}^2 + (1)/(2)3m{v_2'}^2\\v_0^2 = {v_1'}^2 + 3{v_2'}^2`

We have two equations and two unknowns:

`v_0 = v_1' + 3v_2'\\v_0^2 = {v_1'}^2 + 3{v_2'}^2\\\\3v_2' = v_0 - v_1'\\3{v_2'}^2 = {v_0}^2 - {v_1'}^2\\\\3{v_2'}^2 = (v_0 - v_1')(v_0 + v_1') = 3{v_2}'(v_1' + v_0)\\\\v_2' = v_1' + v_0\\3v_2' = v_0 - v_1'\\\\4v_2' = 2v_0\\\\v_2' = v_0/2\\v_1' = -v_0/2`

Step-by-step explanation:

The first cart hits the second cart at rest and turns back with half its speed.

The second cart starts moving to the right with half the initial speed of the first cart.