# Impulse Units

## What is Impulse?

The term Impulse is used to refer to fast acting force or “impact”, Thus impulse can be defined as “The sudden force acting on an object for short interval of time”. Conventionally represented by $J$. Sometimes by $imp$ and expressed in $N.s$

The impulse when plotted against time, the graph of unit impulse is gaussian in nature. the graph looks like- In classical mechanics impulse is expressed as an integral of a resultant force over an interval of time on which it acts. It is a vector quantity. Which implies it has direction along with magnitude. We know that force is a vector quantity, Impulse arise as a result of force, hence this vector quantity follow the same direction as that of force.

Object in the presence of any force, accelerate or change the velocity. The same force acting on the same body for long duration of time will cause greater change in its linear momentum as compared to same for short duration. Similarly, small force applied for a long duration will cause the same change in momentum as a large force - Impulse force for short duration.

Change in momentum is equal to the product of the average force and time duration duration. Mathematically written as

$J=F_{avg}\left ( t_{2}-t_{1} \right )$

We know that Impulse is the integral of force $F$ over time:

$J=\int Fdt$

### Impulse – Momentum Theorem

Impulse $J$ produced between the time interval $t_{1}$ and $t_{2}$ is given by-

$J=\int_{t_{1}}^{t_{2}}Fdt$ —–(1)

Where $F$ is the resultant force acting in the time interval $t_{1}$ and $t_{2}$

Now by applying Newton’s second law of motion, we can relate Force F and momentum P as –

$F=\frac{dP}{dt}$

Thus, substituting it in equation (1) we get-

$J=\int_{t_{1}}^{t_{2}}\frac{dP}{dt}dt$ $\Rightarrow J=\int_{p_{1}}^{p_{2}}dP$ $J=P_{2}-P_{1}$ $J=\Delta P$

“Impulse acting on a body changes equal amount of linear momentum in the same direction.”

This equivalence of Impulse and Momentum is called as Impulse-Momentum Theorem.

Thus, Impulse can be expressed as the change in momentum of an object to which force is applied. When the mass of the object is constant, Then impulse can be written as –

$J=\int_{t_{1}}^{t_{2}}Fdt=\Delta P=mv_{2}-mv_{1}$

Where,

• J is the Impulse
• F is Force applied
• m is the mass of the object
• v1 and v2 are the initial and final velocity respectively during the presence of force
• t1 and t2 are the beginning and end time respectively during the presence of force

## Impulse Units

The units of impulse can be derived using Impulse-momentum theorem. Impulse is equivalent to a change in momentum, This can be mathematically written as-

$\sum\vec{F}\Delta t=m\Delta\vec{v}$

The left hand side of the equation clearly says that Force is multiplied by time to give impulse. According to S.I system unit of Impulse will be newton second or $N.m$

Here newton (N) is a derived unit/compound unit. Using Newton’s second law force can be written as product of mass and square of velocit that is,

$F=ma=mv^{2}$

Expressing above equation in terms of their SI units we get-

$N=kg\frac{m}{s^{2}}$

On multiplying both sides by second we get-

$N.s=kg\frac{m}{s^{2}}.s$ $=kg\frac{m}{s}$ $=kgm/s$

Thus, it has the same units and dimensions as momentum.

 SI unit N.s Dimensional formula [MLT-1] English engineering unit lbf.s Dimensionally equivalent unit of momentum kg.m.s-1

### Real life instances

We come across impulse at various situations of our day to day life. Some of such scenarios are listed below-

• Airbags in cars are designed on the basis of principle of Impulse
• Tossing the coin
• Hitting golf ball
• Batsman hitting the ball during cricket match
• Kick starting the motorbike