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Special Cases of Head-on Elastic Collision

If projectile and target are of same mass, i.e. m1 = m2
 
Since 42405.png
 
and 42398.png.
 
Substituting m1 = m2, we get
 
v1 = u2 and v2 = u1
 
It means when two bodies of equal masses undergo head-on elastic collision, their velocities get interchanged.
 
Example

Collision of two billiard balls
 

42392.png
Fig. 4

 

 
Sub case: u2 = 0 i.e., target is at rest. Therefore, v1 = 0 and v2 = u1
If a massive projectile collides with a light target, i.e., m1 >> m2
 
Since 42451.png
 
and 42444.png
 
Substituting m2 = 0, we get
v1 = u1 and v2 = 2u1 – u2
 
Sub case: u2 = 0, i.e., target is at rest. Therefore, v1 = u1 and v2 = 2u1
 
If a light projectile collides with a very heavy target i.e. m1 << m2
 
Since 42499.png
 
and 42493.png.
 
Substituting m1 = 0, we get
v1 = –u1 + 2u2 and v2 = u2
 
Example

Collision of a ball with a massive wall.
 

42487.png

 

42593.png

Fig. 5

 

 
Sub case: u2 = 0, i.e., target is at rest. Therefore, v1 = – u1 and v2 = 0, i.e., the ball rebounds with same speed in opposite direction when it collides with stationary and very massive wall.




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