Third Law of Motion

The first two laws of motion tell us how an applied force changes the motion and provide us with a method of determining the force. The third law of motion states that when one object exerts a force on another object, the second object instantaneously exerts a force back on the first. These two forces are always equal in magnitude but opposite in direction. These forces act on different objects and never on the same object. In the game of football sometimes we, while looking at the football and trying to kick it with a greater force, collide with a player of the opposite team. Both feel hurt because each applies a force to the other. In other words, there is a pair of forces and not just one force. The two opposing forces are also known as action and reaction forces.

Let us consider two spring balances connected together as shown in Fig.1. The fixed end of balance B is attached with a rigid support, like a wall. When a force is applied through the free end of spring balance A, it is observed that both the spring balances show the same readings on their scales. It means that the force exerted by spring balance A on balance B is equal but opposite in direction to the force exerted by the balance B on balance A. The force which balance A exerts on balance B is called the action and the force of balance B on balance A is called the reaction. This gives us an alternative statement of the third law of motion i.e., to every action there is an equal and opposite reaction. However, it must be remembered that the action and reaction always act on two different objects.

Fig.1.

Suppose you are standing at rest and intend to start walking on a road. You must accelerate, and this requires a force in accordance with the second law of motion. Which is this force? Is it the muscular effort you exert on the road? Is it in the direction we intend to move? No, you push the road below backwards. The road exerts an equal and opposite reaction force on your feet to make you move forward.

It is important to note that even though the action and reaction forces are always equal in magnitude, these forces may not produce accelerations of equal magnitudes. This is because each force acts on a different object that may have a different mass.

When a gun is fired, it exerts a forward force on the bullet. The bullet exerts an equal and opposite reaction force on the gun. This results in the recoil of the gun (Fig. 2). Since the gun has a much greater mass than the bullet, the acceleration of the gun is much less than the acceleration of the bullet. The third law of motion can also be illustrated when a sailor jumps out of a rowing boat. As the sailor jumps forward, the force on the boat moves it backwards.

Fig.2.

Conservation of Momentum

Suppose two objects (two balls A and B, say) of masses m_A and m_B are travelling in the same direction along a straight line at different velocities u_A and u_B, respectively [Fig.3(a)]. And there are no other external unbalanced forces acting on them. Let u_A > u_B and the two balls collide with each other as shown in Fig. 3(b). During collision which lasts for a time t, the ball A exerts a force F_AB on ball B and the ball B exerts a force F_BA on ball A. Suppose v_A and v_B are the velocities of the two balls A and B after the collision, respectively [Fig.3(c)].

Fig.3.

From equation,  p = mν, the momenta (plural of momentum) of ball A before and after the collision are m_Au_A and m_Av_A, respectively. The rate of change of its momentum (or F_AB, action) during the collision will be m_A  (v_A- u_A ) / t .

Similarly, the rate of change of momentum of ball B (= F_BA or reaction) during the collision will be  m_B (v_B-u_B) / t

According to the third law of motion, the force FAB exerted by ball A on ball B (action) and the force F_BA exerted by the ball B on ball
A (reaction) must be equal and opposite to each other. Therefore,

         F_AB = - F_BA                         ------------- (1)

or, m_A (v_A-u_A) / t = - m_B (v_B-u_B) / t
                             
This gives,

m_Au_A + m_Bu_B = m_Av_A + m_Bv_B  ------------- (2)

Since (m_Au_A + m_Bu_B) is the total momentum of the two balls A and B before the collision and (m_Av_A + m_Bv_B) is their total momentum after the collision, from Eq. (2) .We observe that the total momentum of the two balls remains unchanged or conserved provided no other external force acts.

As a result of this ideal collision experiment, we say that the sum of momenta of the two objects before collision is equal to the sum of momenta after the collision provided there is no external unbalanced force acting on them. This is known as the law of conservation of momentum. This statement can alternatively be given as the total momentum of the two objects is unchanged or conserved by the collision.

Summary:

1. Third law of motion: To every action, there is an equal and opposite reaction and they act on two different bodies.

2. In an isolated system, the total momentum remains conserved.