One Dimensional Elastic Collision

Now, we are going to discuss the application of the law of momentum conservation on the collision of two bodies. Collision process may happen very shortly, for example on the collision between two billiard balls, or it may happen in a long time, for example on the collision among stars in heaven. On all collision processes, the colliding bodies will interact with each other strongly only during the collision. If there are external forces, these forces are much smaller compared to the interacting forces and therefore are neglected.

If the total kinetic energy of the colliding bodies after the collision is equal to the total kinetic energy before the collision, the collision process is called completely elastic collision. However, if the total kinetic energy of the two bodies before the collision is not the same with the total kinetic energy after the collision, then the collision process is called inelastic collision. In an inelastic collision, part the of the kinetic energy is changed into another kind of energy, for example, thermal energy. Thus, the total kinetic energy after the collision the two bodies are joined together, the collision process is called completely inelastic collision, as studied in Example 4.2. It should be underlined here that in every collision process where the influence of external forces is neglected, the law of momentum conservation always holds.

1. One-Dimensional Elastic Collision

I have been mentioned that if in a collision there is no lost of kinetic energy, then a collision is elastic. We will discuss a one-dimensional collision, in which the velocities of colliding bodies is on the same straight line, for example along the x-axis.

Figure 4.6 (a) shows two bodies of masses m1 and m2, which are moving with velocities of v1 and v2 along the same straight line. Having collided elastically, their velocities become v1' and v2' (Figure 4.6 (b)). Based on the law of momentum conservation, it is obtained:

2. Two-Dimensional Elastic Collision

If often happens that the motion of colliding bodies is not in a straight line but in the two-dimensional plane. One example of such collisions in the collision between two balls in a billiards game.

Figure 4.7 shows a ball of mass m1 moving along the x-axis is colliding with another ball of mass m2, which in initially at rest. Having colliding, the two ball move into directions that make the angles of Ө₁ and Ө₂ respectively toward the x-axis.

By applying the law of momentum conservation in the x-axis direction, it is obtained

m₁v₁ = m₁v₁’ cos Ө₁ + m₂v₂’ cos Ө₂

 Since initially both bodies are not moving in the direction of the y-axis then the component of momentum in the y-axis directions is zero.

m₁v₁’ sin Ө₁ + m₂v₂’ sin Ө₂ = 0

3. Completely Inelastic Collision

In a completely inelastic collision, after the collision, the two bodies will move together so that it holds v₁’ = v₂’= v'. Thus, the law of momentum conservation becomes:

m₁v₁ + m₂v_{2 = (} m_{1 +} m₂) v’

Therefore, the velocities of the two bodies after the collision can be calculated with the formula. Thus, by measuring the masses and velocities of the bodies before the collision, the velocity after collision could be predicted. In a complete collision, kinetic energy after collision is always smaller than kinetic energy before collision. If kinetic energy before the collision is E_k = ½ m₁V₁'². The kinetic energy after collision is

E_k' = ½ (m₁ + m₂) v’²

4. Inelastic collision

Most collisions between two bodies found in nature are inelastic collisions, for example, a tennis ball colliding with racket or the hit ball in a baseball game. The analysis of inelastic collision involves a quantity called restitution coefficient (e).

Restitution coefficient is defined as the negative value ofthe ratio of relative velocities before and after collision. 

For inelastic collision, the value of restitution coefficient is between zero and one, 0 < e < 1.

Exercise 4.3 

1. A body with the mass of 1 kg is moving to the direction of positive x-axis with a velocity of 2 m/s. The other body which mass is 3 kg, is moving in the opposite direction with a velocity of 2 m/s. After collided, both bodies are moving together as one. Calculate the velocity of both bodies after the collision.
2. Two balls with masses of 3 kg and 6 kg are moving with speeds of 4 m/s and 1 m/s respectively. The two balls are approaching each other along a straight line connecting the center of masses of the balls. After the collision, the ball with a mass of 3 kg stops moving. Determine the value of the restitution coefficient of the two balls.
3. Two bodies A and B have the same masses. Initially, A is moving to the right with an initial velocity of 5 m/s, and after 2 seconds A covers a distance of 14 m. At that time, A and B undergo a completely inelastic collision. If B is initially moving to the left with a velocity of 15 m/s, determine the velocity of both after the collision.

Multiple Choice

1) Andre Agassi is playing tennis using a ball with the mass of 100 g. The ball is approaching him with a velocity of 100 m/s. Andre hits back the ball with a force of 120 N. If the racket touches the ball for 0.02 s, the velocity of the ball is .....
a. 46 kg.m/s
b. 56 kg.m/s
c. 66 kg.m/s
d. 76 kg.m/s
e 86 kg.m/s

2) Among the following moving bodies, the one that experiences the greatest force when the body hits the wall and stops in the same interval of time is ...........
a. 40 kg with a speed of 25 m/s
b. 50 kg with a speed of 15 m/s
c. 100 kg with a speed of 10 m/s
d. 150 kg with a speed of 7 m/s
e. 200 kg with a speed of 5 m/s

3) Someone with the mass of 40 kg is standing on a skateboard, which mass is 2 kg and is moving at a speed of 10 m/s. If he is suddenly jumping forward with the velocity of 4 m/s while the skate board keeps running, the magnitude of the skate board's velocity is ............
a. 120 kg.m/s
b. 130 kg.m/s
c. 135 kg.m/s
d. 140 kg.m/s
e 150 kg.m/s

4) Ball A is moving with momentum mv and colliding with ball B which is moving on the same straight line. If after the collision the momentum of ball A become -3 mv, the increase of the momentum of ball B is ..............
a.  2 mv
b.  -2 mv
c.  3 mv
d.  -4 mv
e.  4 mv

5) A bullet with the mass of 10 g is fired and hits a block which masses 1.49 kg. The block hanging freely on a piece of rope which length is 0.2 m. If the acceleration due to the earth's gravity is 10 m/s² and the rope is displaced by 60 cm measured from its initial position/vertical position, the velocity of the bullet when fired is ........... m/s
a.  300 √3
b.  300 √2
c.  150 √3
d.  50 √2
e.  150

6) A grenade that is initially at rest is suddenly exploding and breaking up into two pieces, which are moving in the opposite directions. The ratio of masses of the two pieces m₁ : m₂ = 1 : 2. If the released energy is 3 x 10⁵ joules, the ratio of the kinetic energy of the first piece and the second one is ..........
a. 1 : 1
b. 2 : 1
c. 1 : 3
d. 5 : 1
e. 7 : 5

7)  A ping pong ball is falling freely from the height of h₁ and reflected back with the lower height of h₂. The restitution coefficient is ..........
a.   e = ( h₂/ h₁ )^1/2
b.   e = ( h₁/ h₂ )^1/2
c.   e = ( h₁/ h₂ )
d.   e = ( h₂/ h₁ )
e.   e = h₁ h₂

Collision Momentum

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