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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 m1 and m2, moving along the same straight line with velocities u1 and u2, respectively. 
The two bodies will collide only if u1 > u2. The difference in initial velocities i.e., (u1 - u2) is called velocity of approach.
Two bodies undergo a head-on collision and continue moving along the same straight line with velocities v1 and v2 in the same direction.
The bodies will separate after collision only if v2 > v1.
The difference in final velocities i.e., (v2 > v1) is called velocity of separation.
According to the law of conservation of momentum,
m1 u1 + m2 u2 = m1 v1 + m2 v2    ... (1)
m1 (u1 - v1) = m2 (v2 - u2)        ... (2)
 
Substituting v2 in eqn. (1), we get,
v subscript 1 equals fraction numerator left parenthesis m subscript 1 space minus space m subscript 2 right parenthesis u subscript 1 plus 2 m subscript 2 u subscript 2 over denominator m subscript 1 space plus space m subscript 2 end fraction
... (3)
• According to the law of conservation of kinetic energy, 
 
1 half m 1 u subscript 1 superscript 2 space plus space 1 half m 2 u subscript 2 superscript 2 space equals space 1 half m 1 v subscript 1 superscript 2 plus 1 half m subscript 2 v subscript 2 superscript 2
m1 (u12 - v12) = m2 (v22 - u22)        ... (4)
 
Dividing eqn. (1) and (2), we get,
u1 - u2 = v2 - v1                          ...(5)
 
From eqn. (3), it follows that in one dimensional elastic collision, the relative velocity approach (u1 - u2) before collision is equal to the relative velocity of separation (v2 - v1) after collision.
Substituting v2 in eqn.(5), we get,     
 
v subscript 2 equals fraction numerator left parenthesis m subscript 2 space minus space m subscript 1 right parenthesis u subscript 2 plus 2 m subscript 1 u subscript 1 over denominator m subscript 1 space plus space m subscript 2 end fraction ... (6)
 
Calculating the final velocities of the two bodies after collision:
 
When the two bodies are of equal masses:
 
Consider that, m1 = m2 = m
From eqn. (3), we get,
 v1 = u2
From eqn. (5),
we get, 
v2 = u1
Thus, if two bodies of equal masses suffer elastic collision in one dimension, then after the collision, the bodies will exchange their velocities.
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