State and Prove Impulse Momentum Theorem with derivation of equation

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In this post, we will State and Prove the Impulse Momentum Theorem with the derivation of the equation. As evident, this theorem or principle is related to impulse and momentum. Hence a prior knowledge of these two will help.
Related study links are provided here: Read about Momentum and here you can read about Impulse as well.

State and Prove Impulse Momentum Theorem with derivation of equation

Impulse Momentum Theorem statement

Impulse momentum theorem states that the change of momentum of a body is equal to the impulse applied to it. Mathematically, its represented with this equation: Δp = F . Δt

Here, Δp = change in momentum. And F . Δt is the impulse applied. Impulse is represented as the product of Applied force F (of considerable amount) and Δt (a very short duration of time when the force is applied)

Here Momentum is the product of mass and velocity of the body and we call it the ‘Inertia to motion’ as well.

And Impulse is produced when a considerable amount of force acts on a body for a very small duration of time. We get its magnitude by multiplying the magnitude of the force by the time duration.

You can quickly read those in detail from the links below and then proceed further for the momentum impulse theorem.

Impulse momentum theorem derivation

Newton’s Second Law of motion states that the rate of change of momentum of an object or a system is proportional to the net force applied on that object or the system.

Again from this Law, we get the definition of force as
F = m a ……………(1)
where a is the acceleration of the body with mass m when a net force F is applied to it.

Now let’s break down this equation:
Let’s say, u and v are the initial and final velocity of the object under acceleration and the time taken for this change of velocity is Δt

F = m a = m (v-u) / Δt

= (mv – mu) / Δt = change in momentum / Δt

so, F = Δp/Δt

i.e., Force = Rate of Change of Momentum

Summation of all force components on a body
= F =  Δp/Δt ………………….(2)

Again we can write,  Δp = F . Δt………….(3)

From equation 3 above we can see that change in momentum is also expressed as the product of force and the time duration.

Now let’s talk about the impulse-momentum theorem.

If we consider the force to be a constant force, then as a definition we state that impulse is the product of the force applied and the time duration.

Impulse (J) = F .Δt ……………(4)

Incorporating expression from equations 3 in equation 4 above we can write,

Impulse(J) = F .Δt = Δp

i.e., Impulse(J) = Change in momentum

Thus the equation of impulse momentum theorem is derived.

So we can see that Impulse is equivalent to the change in momentum(Δp).

We know this equivalence as the impulse-momentum theorem.

This equation shows us how an impulse created by a force can affect the motion of a body.


[Relevant Posts for problem-solving

Impulse Momentum numerical problems set 1 (solved)

Impulse Momentum numerical problems set 2 (solved)]

In the real world, forces are often not constant. Forces may build up from zero over time and also may vary depending on many factors.

Force Time Curve

Finding out the overall effect of all these forces directly would be quite difficult. As we calculate impulse, we multiply the force by time. This is equivalent to finding the area under a force-time curve.

State and Prove Impulse Momentum Theorem with derivation of equation

For variable force, the shape of the force-time curve would be complicated but for a constant force, we will get a simpler rectangle. In any case, the overall net impulse only matters to understand the motion of an object following an impulse.

Impulse Momentum numerical problems set 1 (solved)

Impulse Momentum numerical problems set 2 (solved)

State and Prove Impulse Momentum Theorem with derivation of equation
impulse
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