Impulse Momentum Theorem for JEE

Momentum is a term that has common usage in everyday life. In sports, one gets to hear momentum all the time. People say that the team that has the momentum is on the front foot, and it will take some effort to stop the team from having momentum.  A person that has a lot of momentum in their studies will surely succeed. These are the common uses of the word momentum. It can be inferred from these sentences that someone who is on the move has momentum. 

In physics, one can define momentum as a mass that is in motion. All objects around us have some mass and if the object is in motion or if it's moving then it can be said that it has some momentum. It is now evident that the momentum depends on the mass of the object and its motion or how fast the object is moving. In terms of physics, it can be said that momentum depends on mass and velocity. The mathematical form of momentum and the impulse-momentum theorem will be discussed in detail in this article.

Impulse and Momentum

Momentum in the mathematical form can be written as the mass times the velocity of the object in consideration. 

$p=m\cdot v$

Where p is the momentum, m is the mass, and v is the velocity of the object.

The SI unit of momentum is $\dfrac{kg.m}{s}$. 

Impulse is a term in physics that is used to describe the effect of force acting on a body over the time that it takes to change the momentum of the body. Impulse can be defined as the product of the average net force that acts on an object for a certain duration. The equation for impulse is given as:

$J=F \cdot \Delta t$

The unit of Impulse is the same as that of momentum, ie. $\dfrac{kg.m}{s}$.

Since the units of both the quantities are similar, there has to be some kind of relation between them. This relation is shown by the impulse momentum theorem and is an important part to solve problems involving collisions. 

Impulse Momentum Theorem

We will now state and prove the Impulse Momentum Theorem. The relation between Impulse and momentum can be established by Newton’s second law. According to Newton’s second law,

$F=m\cdot a$

Here F is the net force, m is the mass, and a is the acceleration of the object.

It can be simplified as,

$\begin{align} &F=m \cdot \dfrac{\Delta v}{\Delta t} \\ &F \cdot \Delta t=m \cdot \Delta v \end{align}$

This equation can be called the impulse momentum equation. It can alternately be written as,

$J=m\cdot \Delta v$

The quantity on the left-hand side of the above equation is the impulse and the quantity on the right-hand side of the equation is the change in momentum. This is precisely what the Impulse Momentum principle or theorem states. It states that the impulse exerted is equal to the change seen in the momentum of the object.

If v1 is the initial velocity and v2 is the final velocity then the impulse momentum equation can be written as,

$\begin{align} &J=m \cdot\left(v_{2}-v_{1}\right) \\ &J=m \cdot v_{2}-m \cdot v_{1} \end{align}$

It can further be simplified as,

$J=p_2-p_1$

Here p₂ and p₁ are the final and the initial momenta.

The physics behind collisions is based on the impulse momentum theorem. Collision is the most common application of impulse Suppose a body experiences a force in a collision with another body and the force acts for a particular period of time, this force results in a change in the momentum of the body. The force will either increase the momentum or decrease it, or it can also change its direction without affecting the magnitude. This means that the impulse is responsible for a change in momentum. 

Impulsive Force

We can define impulsive force as the force that acts on a body for a short period of time. Impulsive forces are normally produced during collisions where one of the bodies changes its momentum or velocity. In these cases, impulsive forces are produced because the forces act for a short period of time. In simple words, impulsive forces are forces that act on anybody for a brief interval rather than acting on it steadily for a continuous period of time. The forces that act for a longer time on the bodies are known as non-impulsive forces. One of the classic examples of non-impulsive forces is the gravitational force on the body which is commonly known as weight. 

Conclusion

Momentum is one of the commonly used terms in everyday life. In physics, momentum is defined as the product of the mass and the velocity of any object. It can be said that any object in motion has some momentum. Impulse is the product of the net force and the time that the force acts on the body. The SI units of both momentum and impulse are $\dfrac{kg.m}{s}$ and this suggests that there has to be some connection between them. This connection is stated in the impulse momentum theorem. It states that the impulse is equal to the change in momentum of the body. A force that acts on a body for a short interval of time is known as an impulsive force. The most common example of impulsive force is the force acting on a body during a collision that brings a change in momentum.