Impulse Units

When we hit a punching bag, the punching bag’s momentum changes due to the force applied by our hand for a specific time. This is an impulse. So, we can define impulse as the change in momentum of an object due to a sudden force applied on it for some time. The graph of impulse against time is Gaussian in nature.

Impulse Meaning

Impulse is the change in momentum of an object due to a force applied on it for some time. The graph of impulse against time is Gaussian in nature. We can see that in the diagram below.

Force Versus Time Graph Depicting the Impulse of the Object.

Impulse can be expressed as the integral of resultant force over the interval of time for which it acts. As the force is a vector quantity, Impulse also becomes a vector quantity having direction and magnitude too.

We have seen that impulse is an integral force over time, and the impulse symbol is ‘J’.

Therefore, we can write \[J = \int {Fdt} \].

Impulse Momentum Theorem

Impulse acting on a body is equal to the change in linear momentum.

We saw impulse as \[J = \int {Fdt} \]

Between time interval t1 and t2, we can write

\[J = \int\limits_{{t_1}}^{{t_2}} {Fdt} \]

But \[F = \dfrac{{dp}}{{dt}}\], substituting in above equation, we get,

\[J = \int\limits_{{t_1}}^{{t_2}} {\dfrac{{dp}}{{dt}}dt} \]

This becomes \[J = \int\limits_{{P_1}}^{{P_2}} {dp} \]

which is equal to \[J = {p_2} - {p_1} = \Delta p\].

This is the equation of impulse momentum theorem.

SI Unit of Impulse

Impulse units can be derived from the impulse momentum theorem. Impulse momentum theorem is given as \[J = {p_2} - {p_1} = \Delta p\].

\[\begin{array}{l}p = m\upsilon \\J = \Delta p = m\Delta \upsilon = kgm/s\\\end{array}\]

But

\[\begin{array}{l}kgm/{s^2} = N\\\therefore kgm/s = N.s\end{array}\].

Therefore, the SI unit of J - Impulse becomes N.s and the equivalent unit is kgm/s.

Interesting Facts

• Airbags for safety in our cars are designed on the principle of impulse.

• Tossing the coin is an example of impulse.

• Kick starting a bike is an example of impulse.

Practice Question

What is meant by impulse? Give one example.

Impulse is the change in momentum of an object due to a force applied on it for some time. An example is when we hit a punching bag, punching bag’s momentum changes due to the force applied by our hand for a specific time. This is an impulse.

Solved Problems

1. Consider a free-falling mass of 0.5kg falling from a height h1=7m and is reflected at a height h2 = 3m. Find impulse (take g=10).

Solution: Given,

m= 0.5kg

h1=7m

h2=3m

g=10m/s2

Velocity before collision:

For free falling body \[{\upsilon ^2} = 2gh\]

\[\therefore {\upsilon _0}^2 = 2 \times 10 \times 7 = 140\]

\[\therefore {\upsilon _0} = \sqrt {140} = - 11.8m/s\]

Negative sign indicates direction of object.

Velocity after collision:

\[{\upsilon _t}^2 = {\upsilon _0}^2 + 2gh\]

\[0 = {\upsilon _0}^2 + (2 \times - 10 \times 3)\]

\[{\upsilon _0}^2 = 60\]

\[{\upsilon _0} = \sqrt {60 = } 7.74m/s\]

We know that \[J = {p_2} - {p_1} = \Delta p\].

\[\therefore J = m({\upsilon _t} - {\upsilon _0}) = 0.5 \times (7.74 + 11.8) = 9.77N.s\]

2. Consider a free-falling ball of mass 0.5kg strikes the floor. Velocity before the collision is 5m/s and velocity after the collision is 3m/s. Find impulse.

Solution: Given,

m=0.5kg

V0=-5m/s

Vt=3m/s

We know that \[J = {p_2} - {p_1} = \Delta p\]

\[\therefore J = m({\upsilon _t} - {\upsilon _0}) = 0.5 \times (3 - ( - 5)) = 4N.s\]

Summary

Impulse is the change in momentum of an object due to a sudden force applied on it for some time. The SI unit of impulse is N.s. Impulse is shown by the symbol ‘J’. Impulse is a vector quantity; it has direction and it also has magnitude. Punching a punching bag is a real-life example of impulse.