Name: Work Done on a Block
Uploaded: 2019-07-19
Duration: 3 min 57 s
Description: Check out this solved illustration based on calculating work done on an object by forces acting at different angles. Just click this video to explore!

Question

Explain elastic and inelastic collision. Obtain equations governing elastic collision in one-dimension and solve for final velocities of two colliding particles considering one particle to be at rest before collision.

Accepted Answer

Elastic Collision

A collision between the molecules of a gas is such that there is no loss of kinetic energy. Also, the kinetic energy and the momentum remains conserved. Such type of collision is called the elastic collision.

Molecules of an ideal gas possess an elastic collision. A gas comprises a huge number of molecules, which are perfectly elastic spheres. When these spherical molecules of gas come in a state of Brownian or random motion, they move in different directions with different speeds, ranging from zero to infinity, and obey Newton’s laws of motion.

Also, the size of these molecules is lesser than the distance between them, that’s why the volume occupied by each is considered negligible when compared with the gas volume.

The collision of these molecules with themselves and with the walls of the container is elastic such that their momentum and kinetic energy remains conserved.

Inelastic Collision

We observed the collision at a microscopic level for the ideal gas. However, at the macroscopic level, we can observe inelastic collisions because the energy doesn’t remain conserved. Let’s take an example of inelastic collisions.

Let’s suppose that Rita is driving car at a high-speed, suddenly another car came on the way, and the other car got crashed. Now what happens next is, some kinetic energy of each vehicle gets transformed to sound, heat, or deformation of the parts of the car.

So, when the car hits the first car, the two vehicles collided with each other, and therefore, their kinetic energy didn’t remain conserved, it got changed to another form. Such type of collision in an inelastic collision.

Expression for velocities :-

Let two bodies of masses $m_{1}$ and $m_{2}$ .

Let the velocity of first body is $u_{1}$ and $u_{2}$ the same straight line.

And consider the two bodies collide and after collision $v_{1}$ and $v_{2}$ be the velocities of two masses.

Before collision

Momentum of mass $m_{1}$ $= m_{1} u_{1}$

Momentum of mass $m_{2}$ $= m_{2} u_{2}$

Total momentum before collision

$p_{1} = m_{1} u_{1} + m_{2} u_{2} $

Kinetic energy of mass $m_{1}$ $= \frac{1}{2} m_{1} {u_{1}}^{2}$

Kinetic energy of mass $m_{2}$ $= \frac{1}{2} m_{2} {u_{2}}^{2}$

Thus,

Total kinetic energy before collision is

$K_{i} = \frac{1}{2} m_{1} {u_{1}}^{2} + \frac{1}{2} m_{2} {u_{2}}^{2}$

After collision

Momentum of mass $m_{1}$ $= \frac{1}{2} m_{1} {v_{1}}^{2}$

Momentum of mass $m_{2}$ $= \frac{1}{2} m_{1} {v_{1}}^{2}$

Total momentum after collision

$p_{2} = m_{1} v_{1} + m_{2} v_{2} $

Total kinetic energy after collision

$K_{f} = \frac{1}{2} m_{1} {v_{1}}^{2} + \frac{1}{2} m_{2} {v_{2}}^{2}$

So, according to the law of conservation of momentum

$m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2} \Rightarrow m_{1} (u_{1} - v_{1}) = m_{2} (v_{2} - u_{2}) . . . . . . . . . . . . (1)$

And according to the law of conservation of kinetic energy

$\frac{1}{2} m_{1} {u_{1}}^{2} + \frac{1}{2} m_{2} {u_{2}}^{2} = \frac{1}{2} m_{1} {v_{1}}^{2} + \frac{1}{2} m_{2} {v_{2}}^{2} \Rightarrow m_{1} ({u_{1}}^{2} - {v_{1}}^{2}) = m_{2} ({v_{2}}^{2} - {v_{1}}^{2}) . . . . . . . . . . . . . (2)$

Now dividing the equation

$(u_{1} + v_{1}) = (v_{2} + u_{2}) \Rightarrow u_{1} - u_{2} = v_{2} - v_{1} $

Therefore, this is, relative velocity of approach is equal to relative velocity of separation.

When the second body is at rest, $u_{2} = 0$

$\Rightarrow u_{1} = v_{2} - v_{1}$

Which is required expression.

Explain elastic and inelastic collision. Obtain equations governing elastic collision in one-dimension and solve for final velocities of two colliding particles considering one particle to be at rest before collision.

Important Questions on Laws of Motion

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