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Velocity after collision

Let two bodies A and B collide inelastically and coefficient of restitution is e.
 
where 40601.png
 

From the law of conservation of linear momentum,
 

 
By solving (20) and (21), we get
40594.png
 
Similarly,
40588.png
 
By substituting e = 1, we get the value of v1 and u2 for perfectly elastic head-on collision.

Ratio of velocities after inelastic collision

A sphere of mass m moving with velocity u hits inelastically with another stationary sphere of same mass (Fig. 7).
 
40582.png
Fig. 7
 
40576.png
 
By conservation of momentum:
 
Momentum before collision = Momentum after collision
 
 
Solving (22) and (23), we get 40570.png and 40564.png.
 
∴ 40558.png

Loss in kinetic energy

Loss (ΔK) = Total initial kinetic energy – Total final kinetic energy
40552.png
 
Substituting the value of v1 and v2 from the above expression,
Loss (ΔK) = 40546.png
 
By substituting e = 1, we get ΔK = 0, i.e., for perfectly elastic collision, loss of kinetic energy will be zero or kinetic energy remains constant before and after the collision.




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