• A seasoned cricketer catches a cricket ball coming in with great speed far more easily than a novice, who can hurt his hands in the character of momentum has not been evident. In the examples so far, momentum and change in momentum both have the same direction. But this is not always the case. Suppose a stone is rotated with uniform speed in a horizontal plane by means of a string, the magnitude of momentum is fixed, but its direction changes (Fig.
. ). A force is needed to cause this change in momentum vector. This force is provided by our hand through the string.
Experience suggests that our hand needs to exert a greater force if the stone is rotated at greater speed or in a circle of smaller radius, or both. This corresponds to greater acceleration or equivalently a greater rate of change in momentum vector. This suggests that the greater the rate of change in momentum vector the greater is the force applied. Fig.
. Force is necessary for changing the direction of momentum, even if its magnitude is constant. We can feel this while rotating a stone in a horizontal circle with uniform speed by means of a string. These qualitative observations lead to the second law of motion expressed by Newton as follows : The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts .
Thus, if under the action of a force F for time interval ∆ t , the velocity of a body of mass m changes from v to v + ∆ v i.e. its initial momentum p = m v changes by ∆ ∆ v . According to the Second Law, or k ∆ ∆ ∝ ∆ ∆ where k is a constant of proportionality. Taking the limit ∆ t → , the term ∆ ∆ p becomes the derivative or differential co-efficient of p with respect to t , denoted by d d t p .
Thus k ( . ) For a body