Collision between two bodies - (For example two balls)
If no external force is acting on the system that is two ball, during whole process of colliosion (before the collision, during the collision, and after the collision) the momentum of the two-body system will remain constant.
If before the collision velocities are v1 and v2, masses m1 and m2
m1v1 + m2v2 = P
If during the collsion (at the instant) both have the same velocity V
(m1 + m2)*V = P
After the collision velocities are say v1' and v2'
m1v1' + m2v2' = P
In the case of energy of the system, assuming that there is no friction, the sum of the kinetic energy before the collision and after the collision will be same in case of elastic collision.
One example of elastic collision given in the book is a spring attached to the ball at the beginning which is getting hit from behind. due to the collision, the spring gets compressed and then expands.
Other example is balls perfectly elastic material. In the case of balls, there will be a deformation due to the collision and the surfaces will be in contact for some interval of time and travel together at the same speed. As the forces develop inside them the balls try to regain their original shape, and in the process push each other. The velocity of the front ball increases while that of the rear ball decreases and the balls separate. After the separation, the balls regain their original shape so that the elastic potential energy is completely converted back into kinetic energy.
Thus during an elastic collision, the initial kinetic energy of the two body system is equal to the final kinetic energy, but it does not remain constant as during the collision some kinetic energy is converted into elastic potential energy.
In the case of inelastic collsion, the two deformed balls have no tendency to regain the shape and tend to remain in contact after the deformation and move with same speed. The kinetic energy of the two body system decreases at the time of deformation and remains constant thereafter.
Collisions – Psat JEE Problems
1. A ball hits the floor and rebounds after an inelastic collision. In this case
a. the momentum of the ball just after the collision is the same as that just before the collision.
b. the mechanical energy of the ball remains the same in the collision.
c. the total momentum of the ball and the earth is conserved.
d. the total energy of the ball and the earth is conserved.
(More than one alternative may be correct)
Answer ( c ) and (d)
Note: total kinetic energy is not conserved but total energy is conserved.
2. A particle of mass 4m which is at rest explodes into three fragments. Two of the fragments, each of mass m, are found to move with a speed v, each in mutually perpendicular directions. The total energy released in the process of explosion is _________ .
As the two fragments went in mutual perpendicular directions with velocity v the resultant momentum
2mv cos 45° = 2mv(1/√2) = √2 mv
if the velocity of fragment with 2m mass is V
then 2 mV =/√2mv
V = v/√2
Hence energy released = ½ mv² + ½ mv² + ½ *2 m (v/√2) ²