# Newton's Second Law of Motion

## Table of Contents

Newton's second Law of Motion states that The rate of change of momentum is directly proportional to the force applied in the direction of force.

For example; when acceleration is applied on a moving vehicle, the momentum of the vehicle increases and the increase is in the direction of motion because the force is being applied in the direction of motion. On the other hand, when brake is applied on the moving vehicle, the momentum of the vehicle decreases and the decrease is in the opposite direction of motion because the force is being applied in the opposite direction of motion.

## Mathematical formulation of Newton’s Second Law of Motion:

Let mass of an moving object = `m`.

Let the velocity of the object changes from `'u'`; to `'v'` in the interval of time `'t'`

This means,

Initial velocity of the object = `u`.

Final velocity of the object = `v`.

We know that momentum (p) = Mass x velocity

Therefore,

Momentum (p) of the object at its initial velocity `u = m xx u = m\u`

Momentum (p) of the object at its final velocity `v = m xx v = mv`

The change in momentum `= mv - m\u`

Rate of change of momentum `=(mv-m\u)/t`-----(i)

According to the Newton’s Second Law of motion force is directly proportional to the rate of change of momentum.

This means, Force ∝ Rate of change of moentum

After substituting the value of rate of change of momentum from equation (i) we get.

Force (F) `prop (mv-m\u)/t`

`=>F prop (m(v-u))/t`

`=>F prop m(v-u)/t`

`=>F prop m a`

[∵ acceleration (`a`)`=(v-u)/t`]

[Since, acceleration is the rate of change in velocity]

`=>F=k*m*a` ----(ii)

Where `k` is the proportionality constant

∵ 1 unit force is defined as the mass of 1kg object produces the acceleration of 1m/s^{2}

∴ 1 unit of force `=kxx1 kg xx 1m//s^2`

∴ by putting the value of `k=1` in equation (ii)

`F = m*a`----(iii)

⇒ Force = mass x acceleration

Thus Newton’s Second Law of Motion gives the relation between force, mass and acceleration of an object.

According to the relation obtained above, Newton’s Second Law can be modified as follows:

The product of mass and acceleration is the force acting on the object.

## The SI unit of Force: Newton (N)

Since Force = Mass x Acceleration

The unit of mass = kg and The unit of acceleration = m/s^{2}

If force, mass and acceleration is taken as 1 unit.

Therefore,

1 Newton (N) = 1kg x 1m/s^{2}

Thus, Newton (N) = kg m/s^{2}

Equation (v) can be also written as

`=>a=F/m`

This equation is the form of Newton’s Second Law of Motion. According to this equation, Newton’s Second Law of Motion can also be stated as follow:

The acceleration produced by a moving body is directly proportional to the force applied over it and inversely proportional to the mass of the object.

From the above relation it is clear that

Acceleration increases with increase in force and vice versa.

Acceleration decreases with increase in mass and vice versa.

That’s why a small vehicle requires less force to attain more acceleration while a heavy vehicle requires more force to get the same acceleration.

## Newton’s Second Law of Motion in everyday life:

- A fielder pulls his hand backward; while catching a cricket ball coming with a great speed, to reduce the momentum of the ball with a little delay. According to Newton’s Second Law of Motion; rate of change of momentum is directly proportional to the force applied in the direction. While catching a cricket ball the momentum of ball is reduced to zero when it is stopped after coming in the hands of fielder. If the ball is stopped suddenly, its momentum will be reduced to zero instantly. The rate of change in momentum is very quick and as a result, the player’s hand may get injured. Therefore, by pulling the hand backward a fielder gives more time to the change of momentum to become zero. This prevents the hands of fielder from getting hurt.
- For athletes of long and high jump sand bed or cushioned bed is provided to allow a delayed change of momentum to zero because of jumping of athlete. When an athlete falls on the ground after performing a high or long jump, the momentum because of the velocity and mass of the athlete is reduced to zero. If the momentum of an athlete will be reduced to zero instantly, the force because of momentum may hurt the player. By providing a cushioned bed, the reduction of the momentum of the athlete to zero is delayed. This prevents the athlete from getting hurt.
- Seat belts in car - Seat belts in the vehicles prevent the passenger from getting thrown in the direction of motion. In case of emergency, such as accidents or sudden braking, passengers may be thrown in the direction of motion of vehicle and may get fatal injuries. The stretchable seat belts increase the time of the rate of momentum to be reduced to zero. The delayed reduction of momentum to zero prevents passengers from such fatal injury.