Question

A sphere A impinges directly on an identical sphere B at rest. If $e$ is the coefficient of restitution then determine the ratio of the velocities of A and B after impact.

Answer 1

In the solution we will be using the equation of conservation of momentum which says that momentum of two colliding bodies before collision is equal to their momentum after collision. In other words, it can also be said that the momentum is conserved.

Complete step by step solution:
It is given that sphere A is identical to sphere B which means that the mass of sphere A is equal to the mass of sphere B which is expressed below.

{m_1} = {m_2}\\\ = m \end{array}$$ Let us consider that sphere A is moving with velocity v before collision. Also it can be said that before collision velocity of sphere B is zero. Using the equation of conservation of momentum which states that the momentum of sphere A and sphere B before collision is equal to the momentum of sphere A and sphere B after collision. $$m{u_1} + m{u_2} = m{v_1} + m{v_2}$$……(1) Here m is the mass of sphere A and sphere B, $${u_1}$$ is the velocity of sphere A before collision, $${u_2}$$ is the velocity of sphere B before collision, $${v_1}$$ is the velocity of sphere A after collision and $${v_2}$$ is the velocity of sphere B after collision. Substitute $$0$$ for $${u_2}$$ in equation (1).

m{u_1} + m \cdot 0 = m{v_1} + m{v_2}\\
{u_1} = {v_1} + {v_2}

Write the expression for coefficient of restitution. e = (relative separation velocity)/(relative_impact_velocity).....(3) The expression for the relative velocity of separation. $${\rm}$$ relative separation velocity =$${v_2} - {v_1}$$ The expression for the relative impact velocity. $${\rm}$$ relative impact velocity = $${{\rm{u}}_1}$$ Substitute $${v_2} - {v_1}$$ for $${\rm}$$ relative separation velocity and $${v_1}$$ for relative impact velocity in equation (3). $$\ e = \dfrac{{{v_2} - {v_1}}}{{{u_1}}}\\\ {v_2} - e{u_1} = {v_1} $$……(4) Substitute $${v_2} - e{u_1}$$ for $${v_1}$$ in equation (2).

{u_1} = {v_2} - e{u_1} + {v_2}\\
{v_2} = \dfrac{{\left( {1 + e} \right){u_1}}}{2}

Substitute $$\dfrac{{\left( {1 + e} \right){u_1}}}{2}$$ for $${v_2}$$ in equation (2).

{u_1} = {v_1} + \dfrac{{\left( {1 + e} \right){u_1}}}{2}\\
{v_1} = \dfrac{{\left( {1 - e} \right){u_1}}}{2}

The ratio of velocity of sphere A to the velocity of sphere B after impact. $$R = \dfrac{{{v_1}}}{{{v_2}}}$$……(5) Substitute $$\dfrac{{\left( {1 - e} \right){u_1}}}{2}$$ for $${v_1}$$ and $$\dfrac{{\left( {1 + e} \right){u_1}}}{2}$$ for $${v_2}$$ in equation (5).

R = \dfrac{{\dfrac{{\left( {1 - e} \right){u_1}}}{2}}}{{\dfrac{{\left( {1 + e} \right){u_1}}}{2}}}\\
= \dfrac{{\left( {1 - e} \right)}}{{\left( {1 + e} \right)}}

**Therefore, the ratio of velocities of sphere A to the velocity of sphere B is $$\dfrac{{\left( {1 - e} \right)}}{{\left( {1 + e} \right)}}$$.** **Note:** We can remember the expression for coefficient of restitution to solve similar kinds of problems as represented by equation (3). Also, the law of conservation of momentum is valid when the collision is elastic in nature. The momentum of sphere A and sphere B before collision is equal to the momentum of sphere A and sphere B after collision.