# Second Law of Motion

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## Second Law of Motion

The rate of change of momentum of a moving body is proportional to and is in the same direction as the force acting on it

### F = d(mv)/dt

where F is the applied force, v is the velocity of the body, and m its mass.

If mass remains constant

or

### F =ma

where a is the acceleration The direction of acceleration is the same as that of F.

As F = ma it follows that for a fixed mass, greater the force, greater the acceleration.

The SI unit of force is Newton. 1 Newton is equal to 1 kgm/sec2.

1 Newton is force which will cause an acceleration of 1 m/sec2 in a body of mass 1 kg.

How do we explain the behaviour of a ball which is hit by a bat instead of being thrown.

In the case of a bat the force acts for an extremely short duration. The effect of this force is called an impulse.

Which gives rise to the question. What is impulse?

## What is impulse?

Impulse (J) is the product of a force F and the time t for which it acts.

If the force is variable, the impulse is the integral of Fdt from to to t1. The impulse of a force acting for a given interval is equal to the change in momentum produced over the interval.

### J =m(v1 - v0),

assuming that mass (m) remains constant while the velocity changes from v0 to v1.

## Solved Examples on Second Law of Motion

A bullet of mass 0.04 kg moving with a speed of 90 m/s enters a heavy wooden block and is stopped after it penetrates 60 cm into the block. What is the average resistive force exerted by the block on the bullet?

u = 90 m/s, v = 0, S = 60 cm = 0.6m

v2 - u2 = 2as

-902 = 2 x a x 0.6

a = -902 / 2 x 0.6 = -6750 m/s2

retarding force = m a = 0.04 x 6750 = 270 N

A base ball of mass 200 g moving with a velocity of 15 m/s is brought to rest by a player in 0.05 second. What is the impulse of the ball and the average force applied by the player?

m = 0.2 kg, u = 15 m/s, v = 0

Change in momentum = m(v - u) = -0.2 x 15 = -3 Ns

As change in momentum = impulse = -3 Ns

Impulse(J) = average force(F) x time(t)

F = J/t = 3/0.05 = 60 N