When two tennis balls collide then, after collision, they will rebound with velocities less than the velocities before the impact. During this process, a portion of K.E is lost, partly due to friction as the molecules in the ball move past one another when the balls distort and partly due to its change into heat and sound energies.

A collision in which the K.E of the system is not conserved, is called the inelastic collision.

Under certain special conditions no kinetic energy is lost in the collision.

In the ideal case when no K.E is lost, the collision is said to be perfectly elastic.

For example, when a hard ball is dropped onto a marble floor, it rebounds to very nearly the initial height. It looses negligible amount of energy in the collision with the floor.

It is to be noted that momentum and total energy are conserved in all types of collisions. However, the K.E. is conserved only in elastic collisions.

 before_collision_after_collision

Fig 3.9

Elastic Collision One Dimension

Consider two smooth, non-rotating balls of masses m1 and m2, moving initially with velocities v1 and v2 respectively, in the same direction. They collide and after collision, they move along the same straight line without rotation. Let their velocities after the collision be v’1 and v’2 respectively, as shown in Fig. 3.9.

We take positive direction of the velocity and momentum to the right. By applying the law of conservation of momentum we have

 elastic_collision_in_one_dimension_01

As the collision is elastic, so the K.E is also conserved. From the conservation of K.E we have

elastic_collision_in_one_dimension_02

We note that , before collision elastic_collision_in_one_dimension_03 is the velocity of first ball relative to the second ball. Similarly elastic_collision_in_one_dimension_04 is the velocity of the first ball relative to the second ball after collision. It means that relative velocities before and after the collision has the same magnitude but are reversed after the collision. In other words, the magnitude of relative velocity of approach is equal to the magnitude of relative velocity of separation.

In equations 3.14 and 3.16, m1, m2, v1 and v2 are known quantities. We solve these equations to find the values of v’1 and v’2, which are unknown. The results are

  elastic_collision_in_one_dimension_05

ball_before_collision_and_after_collision

Fig 3.10

There are some cases of special interest, which are discussed below:

elastic_collision_in_one_dimension_06

In this case the mass m2 be at rest, then v 2 = 0 the equations 3.17 and 3.18 give

 elastic_collision_in_one_dimension_07

When m1 = m2 then ball of mass m1 after collision will come to a stop and m2 will take off with the velocity that m1 originally has, as shown in fig 3.11. thus when a billiard ball m1, moving on a table collides with exactly similar ball m2 at rest, the ball m1 stops while m2  begins to move with the same velocity, with which m1 was moving initially.

 ball_ofmass

Fig 3.11

(iii)               When a light body collides with a massive body at rest

In this case initial velocity v2 = 0 and m2 >> m1. Under these conditions m1 can be neglected as compared to m2. From equations 3.17 and 3.18 we have v’1 = -v1 and v’2 = 0

The result is shown in Fig 3.12. this means that m1 will bounce back with the same velocity while m2 will remain stationary. This fact is made used of by the squash player.

case_initial_velocity

Fig 3.12

(iv)              When a massive body collides with light stationary body

In  this case m1 >> m2 and v2 = 0 so m2 can be neglected in equations 3.17 and 3.18. this gives v’1 _ v1 and v’2 _ 2 v1. Thus after the collision, there is practically no change in the velocity of the massive body, but the lighter one bounces off in the forward direction with approximately twice the velocity of the incident body, as shown in Fig. 3.13.

Thus_after_the_collision

Fig 3.13

Example 3.4: A 70 g ball collides with another ball of mass 140 g. the initial velocity of the first ball is 9 ms-1 to the right while the second ball is at rest. If the collision were perfectly elastic what would be the velocity of the two balls after the collision?

Solution:

elastic_collision_in_one_dimension_09

Example 3.5: A 100 g golf ball is moving to the right with a velocity of 20 ms-1. It makes a head on collision with an 8 kg steel ball, initially at rest. Compute velocities of the balls after collision.

Solution: we know that

elastic_collision_in_one_dimension_10

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If_another_car_crashes

If another car crashes into back of yours, the head-rest of the car seat can save you from serious neck injury. It helps to accelerate your head forward with the same rate as the rest of your body.