Conservation of Momentum

Introduction

Do you ever realize that it is really difficult to catch a fast moving ball rather than a slow moving ball? Do you ever think that a fast moving car can cause more damage when met with an accident rather than slow moving car?

This is all because a physical quantity is involved in both cases i.e. momentum. Momentum of an object is the product of mass and velocity of that object. Momentum is a vector quantity and its direction will be along the direction of velocity of the body. So in the case of a fast moving ball and car, both have high velocity so the momentum will be high.

Now what if I say that this physical quantity i.e., momentum of a system will remain conserved in all the cases irrespective of any condition, provided that there is no external force acting on that system. Now you may think - What is the meaning of system here? So the system is the part of this universe within which we are defining our studies.

For example, if two balls are colliding with each other and we have to define the conservation of momentum in this case then these two balls are our system.

Relation between Newton’s Second Law of Motion and Conservation of Momentum

Did you notice that one of the main criteria mentioned above for the conservation of momentum is that there should not be any external force applied on the system? Now from where we are concluding this criteria? What if I say that this criterion comes from Newton's Second Law of motion? Please recall the statement of Newton’s Second Law of Motion, it states that the rate of change of momentum of a body is directly proportional to the force and takes place in the same direction as the applied force. So if there is no external force applied on the system then there is no change in momentum and the momentum remains conserved.

Mathematical derivation

Now let us derive the mathematical expression for the conservation of momentum by taking two balls moving in the same direction and in one-dimension as a system. Let the mass of the balls be $\mathrm{m_1}$ and $\mathrm{m_2}$ respectively. Also, the velocity of mass m1 be $\mathrm{u_1}$ in positive x-axis and the velocity of $\mathrm{m_2}$ be $\mathrm{u_2}$ in the positive x-direction also, as shown in the figure given below [case (a)].

Let us assume that the magnitude of $\mathrm{u_1}$ is greater than the magnitude of $\mathrm{u_2}$. So if they are moving in one dimension then they will collide with each other after some time as shown in the figure [case (b)].

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Figure 1: Co-ordinate axes

During the collision, both the balls will apply the force on each other in opposite directions and since both the forces are acting on each other for a very small span of time, we can say that the balls are exerting impulse on each other during collision. Now you may get confused that there is a force or impulse acting, so how can we conserve the momentum? So the answer is that this force is acting within the system so this force is internal force for the system not the external force.

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Figure 2: Represents all the cases of collision in one dimension

Now let us assume that the velocity after the collision of m₁ and m₂ will be v₁ and v₂, respectively. So if we write the expression for total momentum (initial momentum = pi) before the collision then it will be -

$$\mathrm{p_i = m_1u_1 + m_2u_2 …. (1)}$$

And the momentum (final momentum = pf) after the collision will be -

$$\mathrm{p_f = m_1v_1 + m_2v_2 …. (2)}$$

Now you may ask that the momentum is a vector quantity then how can we add them algebraically? So the answer is that, since both the balls are moving in the same direction and in one-dimension, the velocity in both the cases are in the same direction and we can add them easily by using algebra.

Now as we have seen that when both the balls are colliding, then they will exert a force or impulse on each other but in opposite directions. Let the force exerted by $\mathrm{m_1}$ be $\mathrm{F_{21}}$ and the force exerted by $\mathrm{m_2}$ be $\mathrm{F_{12}}$. Now according to Newton’s Third Law of Motion, both the impulsive force will be equal and opposite, which can be written as -

$$\mathrm{F_{21} = − F_{12}\:\:\:…. (3)}$$

The negative sign is showing that the direction of impulsive force is opposite to each other. Let the time of contact of collision is ‘t’ then according to the Newton’s Second law of motion for the first ball is -

$$\mathrm{F_{21}\:=\:\frac{m_1v_1\:-\:m_1u_1}{t}\:\:\:…. (4)}$$

$$\mathrm{F_{12}\:=\:\frac{m_2v_2\:-\:m_2u_2}{t}\:\:\:…. (5)}$$

Now according to the equation (3) -

$$\mathrm{\Rightarrow\:\lgroup\frac{m_1v_1\:-\:m_1u_1}{t}\rgroup\:=\:-\lgroup\frac{m_2v_2\:-\:m_2u_2}{t}\rgroup}$$

So,

$$\mathrm{\Rightarrow\:m_1v_1\:-\:m_1u_1\:=\:m_2v_2\:-\:m_2u_2}$$

$$\mathrm{\Rightarrow\:m_1u_1\:+\:m_2u_2\:=\:m_1v_1\:+\:m_2v_2}$$

Now from the (1) and (2), we can write that

$$\mathrm{\Rightarrow\:p_i\:=\:p_f}$$

So the momentum remains conserved when there is no external force acting on the system.

Examples of Conservation of Momentum

Rocket Propulsion

The rockets have a gas chamber which ejects gas with very high velocity during the starting of propulsion. Since before the propulsion, the rocket and gasses in the camber were stationary so the total momentum was zero initially. But after propulsion the gas acquires the velocity in the downward direction, so the momentum of the gasses is in the downward direction. So to conserve the momentum the rocket will move in the upward direction.

Recoil of a gun

As we know that the gun is having bullets inside it and both of them remain stationary as a system before firing. Once we fire the gun, then the bullet goes out with a very high velocity and acquires some momentum. In the result to conserve the momentum the gun also acquires some velocity in the opposite direction. This phenomenon is called Recoiling of Gun.

FAQs

Q1. Momentum is a scalar quantity or vector quantity?

Ans. Momentum is a vector quantity because it has both magnitude and direction. Also it obeys the vector law of algebra.

Q2. What is the main criteria for the conservation of momentum in a system?

Ans. There should not be any external force acting on the system to conserve the momentum.

Q3. Give few real life examples of the conservation of momentum.

Ans. Recoiling of gun, Rocket propulsion and explosion of bomb are the few real life examples which shows the conservation of momentum.

Q4. The conservation of momentum is based on which law of motion?

Ans. Conservation of momentum is based on Newton’s third law of motion which states that to every action, there is an equal and opposite reaction. It means that all the forces are acting always in pairs and one body cannot apply the force on another body without experiencing some force on itself.

Q5. What is impulsive force?

Ans. When some force is acting for a very short span of time to cause some change in the momentum then it is called an impulsive force.