# Newton's Second Law of Motion

- Mathematical formulation
- SI unit of force
- Second Law of motion in everyday life

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

**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 a 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 time *t*

This means,

Initial velocity of the object = *u*

Final velocity of the object = *v*

We know that momentum (p) = Mass × velocity

Therefore,

Momentum (p) of the object at its initial velocity *u = m × u = m u*

Momentum (p) of the object at its final velocity *v = m × 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 × 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 × Acceleration

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

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

Therefore,

1 Newton (N) = 1kg × 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:

- While catching a ball during a cricket match, the fielder moves his hands backward. This is done to reduce the momentum of the ball. When the fielder catches the ball, the velocity of the ball reduces to zero. By pulling his hands backward, the fielder allows more time for the speed to come down to zero. This helps in reducing the momentum with subsequent reduction in velocity of tha ball.
- If the ball stops suddenly, its momentum will suddenly come down to zero. In that case, the sudden change in momentum is not going to save the fielder from injury.
- Athletes doing high jump and long jump have the facility to land on a soft bed of sand or foam. Sand and foam provide cushion to allow a little bit more time for the momentum to come to zero. This helps the athlete to avoid injury which could happen due to sudden impact.
- Seat belts are very effective safety device. When a car comes to a sudden stop, the passenger has a tendency to be thrown forward because of inertia. The seat belt prevents it by confining the passenger in its limits. Moreover, the seat belt also allows more time for the momentum to come to zero because seat belt is flexible enough to allow some degree of movement.