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Perfectly Elastic Head-on Collision

Let two bodies of masses m1 and m2 moving with initial velocities u1 and u2 in the same direction and they collide such that after collision their final velocities are v1 and v2, respectively (Fig. 2).
 
According to the law of conservation of momentum,
 

 
According to the law of conservation of kinetic energy
 
40980.png
 
Dividing (4) by (2),
 
 
Relative velocity of approach = Relative velocity of separation

 

Notes:
  • The ratio of relative velocity of separation and relative velocity of approach is defined as coefficient of restitution.
     
    41280.png or 41274.png
  • For perfectly elastic collision, e = 1.
     
    ∴ 41268.png  [as shown in Eq. (6)]
  • For perfectly inelastic collision, e = 0.
     
    ∴ 41262.png or 41256.png
     
    It means that two body stick together and move with same velocity.
  • For inelastic collision, 0 < e < 1.
     
    ∴ 41250.png
     
    In short we can say that e is the degree of elasticity of collision and it is dimension-less quantity.
 
Further from (5) we get v2 = v1 + u1 – u2.
 
 
Substituting this value of v2 in (1) and rearranging, we get
 
40943.png
Similarly, we get
 
40937.png




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