Question

Derive an expression of inelastic collision where there is loss of kinetic energy

Answer 1

Collisions occur when one object strikes another. Problems involving collisions are usually solved using conservation of momentum and energy.

It is categorized into two types,

Inelastic collision

Elastic collision

What is Inelastic Collision?

Inelastic collision is such a type of collision that takes place between two objects in which some energy is lost. The special case of inelastic collision is known as perfectly inelastic collision where after collision two objects stick together. Incase of inelastic collision momentum is conserved but the kinetic energy is not conserved. Most of the collisions in daily life are inelastic in nature.

Elastic Collision:

Any collisions in which the collided objects get separated after collision is known as elastic collision. In case of elastic collision kinetic energy gets conserved. One must use both conservation of momentum and conservation of energy to find the motions of the objects later. Some examples of elastic collisions are: Ping-pong balls, billiards, etc.

Formula for Inelastic Collision:

\(V=\frac{(M_{1}V_{1}+M_{2}V_{2})}{(M_{1}+M_{2})}\)

Where,

V= Final velocity

M1= mass of the first object in kgs

M2= mas of the second object in kgs

V1= initial velocity of the first object in m/s

V2= initial velocity of the second object in m/s

Answer 2

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From conservation of momentum

${m}_{1 }{v }_{1}$

$= ({m}_{1} +{m }_{2} ){v}-{2}→{v}_{2}$

$= \frac{{m}_{1}}{{m}_{1} +{m}_{2}}×{ v }_{1}$

The ratio of kinetic energies before & after collision is

$\frac{{KE}_{f}}{{KE}_{i}}$

$= \frac{\frac{1}{2}×({m}_{1}+{m}_{2}) × (\frac{{m}_{1}}{{m}_{1} +{m}_{2}}×{ v }_{1})²}{\frac{1}{2}×{m}_{1}{v²}_{1}}$

$= \frac{{m}_{1}}{{m}_{1} +{m}_{2}}{×}{ v }_{1}$

The fraction of kinetic energy lost is

$\frac{{KE}_{i} – {KE}_{f}}{{KE}_{i}}$

$= \frac{1 –( \frac{{m}_{1}}{{m}_{1} +{m}_{2}})×{ v }_{1}}{{KE}_{i}} × {KE}_{i}$

$= \frac{{m}_{2}}{{m}_{1} +{m}_{2}}×{ v }_{1}$

Hence energy always loss in inelastic collision.

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