Question

Two perfectly elastic particles P and Q of equal mass travelling along the line joining them with velocities \[15{\text{ }}m/sec\] and \[10{\text{ }}m/sec\]. After collision, their velocities respectively (in m/sec) will be.
A. $0,25$
B. $5,20$
C. $10,15$
D. $20,5$

Answer 1

Hint:In this question, we are given that there are two perfectly elastic particles P and Q with the initial velocities \[15{\text{ }}m/sec\] and \[10{\text{ }}m/sec\]. Also, the masses of both the particles are the same. We have to calculate their velocities after collision. To calculate the velocities, we’ll use the formula of velocities after collision (elastic bodies).

Formula used:
When two bodies collides (elastically) then the velocities are;
${v_1} = \dfrac{{\left( {{m_1} - {m_2}} \right){u_1} + 2{m_2}{u_2}}}{{{m_1} + {m_2}}}$, ${v_2} = \dfrac{{\left( {{m_2} - {m_1}} \right){u_2} + 2{m_1}{u_1}}}{{{m_1} + {m_2}}}$
Here, ${m_1},{m_2}$ and ${u_1},{u_2}$ are the masses and initial velocities of both the bodies respectively.

Complete step by step solution:
Given that,
Velocities of particles P and Q are \[15{\text{ }}m/sec\] and \[10{\text{ }}m/sec\] respectively.
Initial velocity of P, ${u_1} = 15{\text{ }}m/sec$
Initial velocity of Q, ${u_2} = 10{\text{ }}m/sec$
The Masses of the particles P and Q are the same.
Therefore, let ${m_1} = {m_2} = m$

After collision their velocities will be,
Using the formula,
${v_1} = \dfrac{{\left( {{m_1} - {m_2}} \right){u_1} + 2{m_2}{u_2}}}{{{m_1} + {m_2}}} \\ $
$\Rightarrow {v_2} = \dfrac{{\left( {{m_2} - {m_1}} \right){u_2} + 2{m_1}{u_1}}}{{{m_1} + {m_2}}} \\ $
Put the value of ${u_1} = 15{\text{ }}m/sec$, ${u_2} = 10{\text{ }}m/sec$ and ${m_1} = {m_2} = m$
It will be, velocity of P, ${v_1} = \dfrac{{\left( {m - m} \right)15 + 2m\left( {10} \right)}}{{m + m}}$
${v_1} = \dfrac{{20m}}{{2m}}$
$\Rightarrow {v_1} = 10{\text{ }}m/sec$
And Velocity of Q,
${v_2} = \dfrac{{\left( {m - m} \right)10 + 2m\left( {15} \right)}}{{m + m}} \\ $
$\Rightarrow {v_2} = \dfrac{{30m}}{{2m}} \\ $
$\therefore {v_2} = 15{\text{ }}m/sec$

Hence, option C is the correct answer.

Note: To solve such problems, one should always remember the concept of collision of elastic bodies that if the mass is the same and the bodies are perfectly elastic then after collision their velocities get interchanged. Consider the two particles or bodies of equal mass moving in opposite directions at the same velocity. In this case, both bodies come to rest after the collision, implying that no velocity exchange occurs.