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. 9.14(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. 9.14(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. 9.14(c)].
From Eq. (9.1), the momenta (plural of momentum) of ball A before and after the collision are `m_A u_A` and `m_A v_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 `F_(AB)` 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)`(9.6)
or ` m_A ( v_A - u_A)/t = m_B ( v_B - u_B)/t`
`m_A u_A + m_B u_B = m_A v_A + m_B v_B` (9.7)
Since `(m_A u_A + m_B u_B)` is the total momentum of the two balls A and B before the collision and `(m_A v_A + m_B v_B)` is their total momentum after the collision, from Eq. (9.7) 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.
Activity ______________ `9.5`
♦ Take a big rubber balloon and inflate it fully. Tie its neck using a thread. Also using adhesive tape, fix a straw on the surface of this balloon.
♦ Pass a thread through the straw and hold one end of the thread in your hand or fix it on the wall.
♦ Ask your friend to hold the other end of the thread or fix it on a wall at some distance. This arrangement is shown in Fig. 9.15.
♦ Now remove the thread tied on the neck of balloon. Let the air escape from the mouth of the balloon.
♦ Observe the direction in which the straw moves.
Activity ______________ `9.6`
♦ Take a test tube of good quality glass material and put a small amount of water in it. Place a stop cork at the mouth of it.
♦ Now suspend the test tube horizontally by two strings or wires as shown in Fig. 9.16.
♦ Heat the test tube with a burner until water vaporises and the cork blows out.
♦ Observe that the test tube recoils in the direction opposite to the direction of the cork.
♦ Also, observe the difference in the velocity the cork appears to have and that of the recoiling test tube.