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Momentum and Impulse: Getting the Impulse Momentum Theorem


Momentum

While playing some sort of body-contact game, if we are pushed by a heavy but slow moving person, we begin to move. Similarly if a lighter person comes with higher velocity and pushes us we may face the same net effect and again begin to move. In physics, these effects can be well explained by a quantity called linear momentum (p). We will discuss this in details here. We will touch upon its definition, its relationship with Newton’s second law of motion and the  Impulse Momentum Theorem.

Definition of Momentum

Definition: Linear Momentum or Momentum is defined as the product of the mass and velocity of an object. It’s designated with sign p.

Formula or equation of Momentum: So as per definition, p = mv where m is the mass of the object with velocity v. SI unit of momentum is Kg meter/second.

From the above example and the equation, we see that p is directly proportional to the mass and velocity of an object. As a result a heavy but slowly moving object may have a p equal (in magnitude) to that of a light but fast moving object.

Momentum is a vector: Because the velocity v is a vector quantity with both magnitude and direction, the momentum (p) too is a Vector quantity. As p is a vector, the total p of a system consisting of objects is the vector sum of the momenta of all the objects in that system.

Newton’s Second Law of Motion and Momentum

Newton’s Second Law of motion states that the rate of change of momentum of an object or a system is proportional to 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 on 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 = Δp/Δt

Summation of F =  Δp/Δt ………………….(2)

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



Impulse and Momentum

If a force acts on a body for a very brief time then we say that an impulse is generated. As example, we can take hitting a ball with a bat for a brief period of time.

Here a considerable amount of force acts on a body for a relatively shorter period of time.

Impulse Momentum theorem:



Now if we consider the force to be a constant force, then as 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
i.e., Impulse(J) =Δp

Impulse is equivalent to the change in momentum(Δp).

This equivalence is known as the impulse-momentum theorem. This equation shows us how an impulse created by a force can affect the motion of a body.

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

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.

Related study:
Angular Momentum
Force

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