Name: Forces: Conservative and Non-conservative
Uploaded: 2019-07-19
Duration: 4 min 29 s
Description: In this video, let us define the conservative and non-conservative forces and determine which forces are conservative and which are not. Learn more.

Question

Prove that in an elastic collision in one dimension, the relative velocity of approach before impact is equal to the relative velocity of separation after impact.

Accepted Answer

Consider two elastic bodies of masses $m_{1}$ and $m_{2}$ moving along a straight line with velocities ${\vec{u}}_{1}$ and $\vec{u_{2}}$

If $u_{1} > u_{2}$ then bodies will collide head-on as shown in fig. $1$ .

As the collision is head-on bodies continues to move along same strength line.

According to law of conservation of lined momentum

$m_{1} \vec{u_{1}} + m_{2} \vec{u_{2}} = m_{1} \vec{v_{1}} + m_{2} \vec{v_{2}}$

As all velocities are in same direction we can take them to be: $u_{1}, u_{2}, v_{1}, v_{2}$

This gives:

$m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$

or $m_{1} (u_{1} - v_{1}) = m_{2} (v_{2} - u_{2})$ ..................(i)

As the collisions are perfectly elastic, kinetic energy of the bodies is separately conserved.

$∴ \frac{1}{2} m_{1} {u^{2}}_{1} + \frac{1}{2} m_{2} {u^{2}}_{2} = \frac{1}{2} m_{2} ({v^{2}}_{2} - {u^{2}}_{2})$

or $m_{1} ({u^{2}}_{1} - {v^{2}}_{1}) = m_{2} ({v^{2}}_{2} - {u^{2}}_{2})$ ..................(ii)

Divide equation (ii) by (i)

$\frac{({u^{2}}_{1} - {v^{2}}_{1})}{u_{1} - v_{1}} = \frac{({u^{2}}_{2} - {u^{2}}_{2})}{v_{2} - u_{2}}$

or $u_{1} + v_{1} = v_{2} + u_{2}$

or $u_{1} - u_{2} = v_{2} - v_{1}$ ......................(iii)

$u_{1} - u_{2}$ is relative velocity of approach and $v_{2} - v_{1} =$ Relative velocity of separation

Hence, relative velocity of approach $=$ Relative velocity of separation.

Prove that in an elastic collision in one dimension, the relative velocity of approach before impact is equal to the relative velocity of separation after impact.

Important Questions on Work, Power and Energy

Learn and Prepare for any exam you want !