# CBSE Class 11-science Answered

**prove that in an elastic one dimensional collision between two bodies, the relative velocity of approach before collision is equal to the relative velocity of seperation after the collision. Hence derive expressions for the velocities of two bodies in terms of their initial velocities before collision.**

Asked by shafiaagha999 | 27 Feb, 2020, 11:40: PM

Expert Answer

• The collision in which both the momentum and kinetic energy are conserved and the colliding bodies continue to move along the same straight line after the collision is called an elastic collision in one dimension.

• Let two perfectly elastic bodies A and B of masses m

_{1}and m_{2}, moving along the same straight line with velocities u_{1}and u_{2}, respectively.• The two bodies will collide only if u

_{1}> u_{2}. The difference in initial velocities i.e., (u_{1}- u_{2}) is called velocity of approach.• Two bodies undergo a head-on collision and continue moving along the same straight line with velocities v

_{1}and v_{2}in the same direction.• The bodies will separate after collision only if v

_{2}> v_{1}.The difference in final velocities i.e., (v

_{2}> v_{1}) is called velocity of separation.• According to the law of conservation of momentum,

m

_{1}u_{1}+ m_{2}u_{2}= m_{1}v_{1}+ m_{2}v_{2}... (1)m

_{1}(u_{1}- v_{1}) = m_{2}(v_{2 }- u_{2}) ... (2)Substituting v

_{2}in eqn. (1), we get,... (3)

• According to the law of conservation of kinetic energy,

m

_{1}(u_{1}^{2}- v_{1}^{2}) = m_{2}(v_{2}^{2}- u_{2}^{2}) ... (4)Dividing eqn. (1) and (2), we get,

u

_{1}- u_{2}= v_{2}- v_{1}...(5)From eqn. (3), it follows that in one dimensional elastic collision, the relative velocity approach (u

_{1}- u_{2}) before collision is equal to the relative velocity of separation (v_{2}- v_{1}) after collision.Substituting v

_{2}in eqn.(5), we get, ... (6)

• Calculating the final velocities of the two bodies after collision:

When the two bodies are of equal masses:

Consider that, m

_{1}= m_{2}= mFrom eqn. (3), we get,

v

From eqn. (5),

_{1}= u_{2}From eqn. (5),

we get,

v

Thus, if two bodies of equal masses suffer elastic collision in one dimension, then after the collision, the bodies will exchange their velocities.

_{2}= u_{1}Thus, if two bodies of equal masses suffer elastic collision in one dimension, then after the collision, the bodies will exchange their velocities.

Answered by Shiwani Sawant | 28 Feb, 2020, 11:08: AM

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