Elastic and Inelastic Collision-Conservation of Momentum-Solved Examples

Law of Conservation of Momentum

From Newton’s law, we know that the time rate change of the momentum of a particle is equal to the net force acting on the particle and is in the direction of that force.

\(\begin{array}{l}F_{net} = \frac{dp}{dt}\end{array} \)

This clearly makes us understand that linear momentum can be changed only by a net external force. If there is no net external force, p cannot change.

\(\begin{array}{l}{{F}_{net}}=0;\frac{dp}{dt}=0\to p~is~constant\end{array} \)

This implies that there is no change in the momentum, or the momentum is conserved, i.e.,

\(\begin{array}{l}initial \;momentum = final\;momentum\end{array} \)

In order to apply the conservation of momentum, you have to choose the system in such a way that the net external force is zero.

Example:

In the example given below, the two cars of masses m₁ and m₂ are moving with velocities v₁ and v₂ before the collision. And their velocities change to

\(\begin{array}{l}v_{1}^{‘}~and~v_{2}^{‘}\end{array} \)

after collision. To apply the law of conservation of linear momentum, you cannot choose any one of the cars as the system. If so, then there is an external force on the car by another car. So we choose both cars as our system of interest. This is why in all collisions, if both the colliding objects are considered as a system, then linear momentum is always conserved (irrespective of the type of collision).

Solved Problem

A bullet of mass m leaves a gun of mass M kept on a smooth horizontal surface. If the speed of the bullet relative to the gun is v, the recoil speed of the gun will be_______________________.

No external force acts on the system (gun + bullet) during their impact (till the bullet leaves the gun). Therefore, the momentum of the system remains constant. Before the impact (gun + bullet) was at rest. Hence, its initial momentum is zero. Therefore, just after the impact, the momentum of the system (gun + bullet will be zero).

\(\begin{array}{l}\Rightarrow M\vec{V}_b+M\vec{V}_g = 0\end{array} \)

\(\begin{array}{l}\Rightarrow m[\vec{V}_{bg}+\vec{V}_g]+M\vec{V}_g = 0\end{array} \)

\(\begin{array}{l}\vec{V}_g = -\frac{m\vec{V}_{bg}}{M+m}, where \left | \vec{V}_{bg} \right | = velocity\;of\;bullet\;relative\;to\;the\;gun = v\end{array} \)

\(\begin{array}{l}\vec{V}_g= – \frac{mv}{M+m}\end{array} \)

Types of Collision

The common normal at the point of contact between the bodies is known as the line of impact. If the mass centres of both the colliding bodies are located on the line of impact, the impact is called central impact, and if the mass centres of both or any one of the colliding bodies are not on the line of impact, the impact is called eccentric impact.

Direct impact/collision

If velocities vectors of the colliding bodies are directed along the line of impact, the impact is called a direct or head-on impact.

Example

Oblique impact/collision

If velocity vectors of both or of any one of the bodies are not along the line of impact, the impact is called an oblique impact.

Example

Consider a ball of mass m₁ moving with velocity v₁ collides with another ball of mass m₂ approaching with velocity v₂. After a collision, they both move away from each other, making an angle with the line of impact.

Coefficient of restitution

When two objects collide head-on, their velocity of separation after impact is in a constant ratio to their velocity of approach before impact.

\(\begin{array}{l}e = \frac{velocity\;of\;separation}{velocity\;of\;approch}\end{array} \)

\(\begin{array}{l}0\le e\le 1\end{array} \)

\(\begin{array}{l}velocity\;of\;separation = e(velocity\;of\;approach)\end{array} \)

The constant e is called the coefficient of restitution.

Elastic Collision

An elastic collision between two objects is one in which total kinetic energy (as well as total momentum) is the same before and after the collision.

Example

On a billiard board, a ball with velocity v collides with another ball at rest. Their velocities are exchanged, as it is an elastic collision.

Inelastic Collision

An inelastic collision is one in which total kinetic energy is not the same before and after the collision (even though momentum is constant).

Examples

When the colliding objects stick together after the collision, as happens when a meteorite collides with the Earth, the collision is called perfectly inelastic.

In other inelastic collisions, the velocities of the objects are reduced, and they move away from each other.

Velocities of Colliding Bodies after Collision in 1- dimension

Let there be two bodies with masses, m₁ and m_2, moving with velocities u₁ and u₂. They collide in an instant and acquire velocities v₁ and v₂ after the collision. Let the coefficient of restitution of the colliding bodies be e. Then, applying Newton’s experimental law and the law of conservation of momentum, we can find the value of velocities v₁ and v₂.

Conserving momentum of the colliding bodies before and the after the collision.

\(\begin{array}{l}m_1u_1+m_2u_2 = m_1v_1+m_2v_2……(i)\end{array} \)

Applying Newton’s experimental law,

\(\begin{array}{l}We\;have\;\frac{v_2-v_1}{u_2-u_1}=-e\end{array} \)

\(\begin{array}{l}v_2 = v_1-e(u_2 – u_1)……(ii)\end{array} \)

Putting (ii) in (i), we obtain

\(\begin{array}{l}m_1u_1+m_2u_2 =m_1v_1+ m_2({v_1-e(u_2 – u_1))}\end{array} \)

\(\begin{array}{l}v_1 = u_1\frac{(m_1-em_2)}{m_1+m_2}+u_2\frac{m_2(1+e)}{m_1+m_2}……(iii)\end{array} \)

When the collision is elastic, e = 1

\(\begin{array}{l}Finally,\;v_1 = \left ( \frac{m_1 – m_2}{m_1+m_2} \right )u_1+\left ( \frac{2m_2}{m_1+m_2} \right )u_2\end{array} \)

Similarly,

\(\begin{array}{l}\;v_2 = \left ( \frac{m_2 – m_1}{m_1+m_2} \right )u_2+\left ( \frac{2m_1}{m_1+m_2} \right )u_1\end{array} \)

Having derived the velocities of colliding objects in a 1-dimensional collision (elastic and inelastic), let us try to set conditions and draw useful conclusions.

Cases:

1. If m₁ = m₂

v₁ = u₂ and v₂ = u₁

When the two bodies of equal mass collide head-on elastically, their velocities are mutually exchanged.

2. If m₁ = m₂ and u₂ = 0, then

v₁ = 0, v₂ = u₁ 

3. If the target particle is massive.

(a) m₂ >> m₁  and u₂ = 0

v₁ = -u₁ and v₂= 0 

The light particle recoils at the same speed while the heavy target remains practically at rest.

(b) m₂ >> m₁ and u₂ ≠ O

v₁ ≈ – u₁+ 2u₂ and v₂ ≈  u₂

4. If the particle is massive: m₁ >> m₂

v₁ = u₁ and v₂= 2u₁—  u₂

If the target is initially at rest, u₂ = O

v₁ = u₁ and v₂ = 2 u₁

The motion of the heavy particle is unaffected, while the light target moves apart at a speed twice that of the particle.

5. When the collision is perfectly inelastic, e = O

\(\begin{array}{l}v_{1} = v_{2} = \frac{u_{1}m_{1}}{m_{1} + m_{2}} + \frac{u_{2}m_{2}}{m_{1} + m_{2}} = \frac{m_{1}u_{1}+m_{2}u_{2}}{m_{1} + m_{2}}\end{array} \)

Elastic Collision in Two Dimension

A particle of mass m₁ moving with velocity v₁ along the x-direction makes an elastic collision with another stationary particle of mass m₂. After the collision, the particles move in different directions with different velocities.

Applying the law of conservation of momentum,

Along x-axis:

\(\begin{array}{l}m_1V_1 = m_1{v_1}’\cos\theta_1+m_2{v_2}’\cos\theta_2\end{array} \)

Along y-axis:

\(\begin{array}{l}0= m_1{v_1}’\sin\theta_1+m_2{v_2}’\sin\theta_2\end{array} \)

Conservation of kinetic energy,

\(\begin{array}{l}\frac{1}{2}m_1{v_1}^{2} = \frac{1}{2}m_1{{v_1}’}^{2}+\frac{1}{2}m_2{{v_2}’}^{2}\end{array} \)

Given the values of

\(\begin{array}{l}\theta_1 and \;\theta_2\end{array} \)

, we can calculate the values of

\(\begin{array}{l}{v_1}’ and \;{v_2}’\end{array} \)

by solving the above examples.

Variable Mass System

Newton’s Second Law for the Rocket

\(\begin{array}{l}F – mg = m\frac{dV}{dt}\end{array} \)

\(\begin{array}{l}\lambda v_{e} – mg = m\frac{dV}{dt}\end{array} \)

\(\begin{array}{l}dV = \frac{\lambda v_{e}}{m} dt- gdt\end{array} \)

Let m be the instantaneous mass and m_o be the initial mass of the rocket.

 ⇒ m = m_o – λt

\(\begin{array}{l}\Rightarrow \int_{o}^{v} dV = \lambda v_{e}\int_{o}^{t}\frac{dt}{m_{o} – \lambda t} – \int_{o}^{t}  gdt\end{array} \)

Assuming g to be uniform,

\(\begin{array}{l}V = v_{e}ln\frac{m_{o}}{m_{o}-\lambda t} – gt\end{array} \)

\(\begin{array}{l}= v_{e}ln \frac{m_{o}}{m} – gt\end{array} \)

Ignoring the effect of gravity, the instantaneous velocity and acceleration of the rocket (attributed only to the thrust by the ejected mass) are

\(\begin{array}{l}V = v_{e}ln \frac{m_{o}}{m}\end{array} \)

Where,

m = m_o – λt ……………..(3)

Rocket Equations

At any instant of time momentum of the mass (λdt) relative to the ground before ejection,

\(\begin{array}{l}=\lambda dt\: V\hat{j}\end{array} \)

\(\begin{array}{l}\text{After ejection} = (\lambda dt)(-v_{e}\hat{j} + V\hat{j})\end{array} \)

Therefore,

\(\begin{array}{l}\text{Change in momentum} = (\lambda dt)v_{e}\hat{j}\end{array} \)

Rate of change of momentum of the ejected mass

\(\begin{array}{l}=-\lambda v_{e}\hat{j}\end{array} \)

Force exerted on the ejected mass

\(\begin{array}{l}=-\lambda v_{e}\hat{j}\end{array} \)

Force exerted on the system (rocket), i.e.,

\(\begin{array}{l}\text{Thrust}\ F = \lambda v_{e}\hat{j}\end{array} \)

Problem

After a perfectly inelastic collision between two identical particles moving at the same speed in different directions, the speed of the particles becomes (2/3)^rd of the initial speed. The angle between the two particles before the collision is

\(\begin{array}{l}(a)\cos^{-1}(-1/3)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\(b)\sin^{-1}(-1/3 )\;\;\;\;\;\; \\ \;\;\;\;\;\;\;\;\\(c)\cos^{-1(-1/9)}\;\;\;\;\;\;\;\;\;\;\;\;\;\\(d)\sin^{-1}(-1/9)\end{array} \)

Let θ be the desired angle. The linear momentum of the system will remain conserved. Hence,

\(\begin{array}{l}P^{2} = P_{1}^{2}+P_{2}^{2}+2P_1P_2\cos\theta\end{array} \)

\(\begin{array}{l}=\left [ 2m\left ( \frac{2}{3} \right )v \right ]^{2}=(mv)^{2}+2(mv)^{2}+2(mv)(mv)\cos\theta\end{array} \)

\(\begin{array}{l}\Rightarrow \frac{16}{9} = 2+2\cos\theta\end{array} \)

\(\begin{array}{l}\Rightarrow 2\cos\theta= \frac{-2}{9}\end{array} \)

\(\begin{array}{l}\Rightarrow \cos\theta= \frac{-1}{9}\end{array} \)

\(\begin{array}{l}\Rightarrow \theta = 96.37^{0}\end{array} \)

Hence, (C) is the correct choice.

Frequently Asked Questions (FAQs) on Conservation of Momentum

Q1

State the law of conservation of momentum.

The law of conservation of momentum states that the momentum of an isolated system before the collision is equal to the momentum of an isolated system after the collision.

Q2

On what principle does a rocket work?

The law of conservation of linear momentum.

Q3

An astronaut in open space is away from his spaceship. How can he return to his ship?

He must throw some object away from his ship. The astronaut moves backwards to conserve the linear momentum (i.e., recoil). This allows him to return to his ship.

Q4

Why does a gun recoil back when it is being fired?

The gun recoils back when it is being fired to conserve linear momentum.