Question

A ball of mass 4 kg moving on a smooth horizontal surface makes an elastic collision with another ball of mass m at rest in the line of motion of first ball, if after collision first ball moves in the same direction with one fourth of its velocity before collision, then mass of second ball is:
(A) 4 kg
(B) 4.4 kg
(C) 2.4 kg
(D) 2 kg

Answer 1

We should know that elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. If a body regains its original configuration immediately and completely after removal of the deforming force, it would be called Perfectly Elastic Body. The materials which show large plastic ranges beyond elastic limits are called ductile materials, such as copper, silver, iron, aluminium and many more. Elasticity measures how stiff or stretchy a material is. One important way to measure elasticity is to calculate the elastic modulus also known as Young's modulus of the material. The elastic modulus or denoted by E is defined as the stress applied to the material divided by the strain. Based on this concept we have to solve this question.

Complete step by step answer
Let us first write the quantities that are mentioned in the question:
${{\text{m}}_{1}}=4\text{kg}$ and ${{\text{u}}_{1}}=\text{u}$ (let us consider) ${{\text{m}}_{2}}=\text{m}$and ${{\text{u}}_{2}}=0$ (when the ball is at the rest)
Also, $\mathrm{v}_{1}=\mathrm{u} / 4$ and collision is elastic.
We should know that collision, also called impact the sudden, forceful coming together in direct contact of two bodies, such as, for example, two billiard balls, a golf club and a ball, a hammer and a nail head, two railroad cars when being coupled together, or a falling object and a floor.
From collision theory we can write the expression that:
$\mathrm{v}_{1}=\left(\dfrac{\mathrm{m}_{1}-\mathrm{m}_{2}}{\mathrm{m}_{1}+\mathrm{m}_{2}}\right) \mathrm{u}_{1}+\left(\dfrac{2 \mathrm{m}_{2}}{\mathrm{m}_{1}+\mathrm{m}_{2}}\right) \mathrm{u}_{2}$
Now we have to put the values in the above expression to get:
$\dfrac{\text{u}}{4}=\left( \dfrac{4-\text{m}}{4+\text{m}} \right)\text{u}+0\Rightarrow 4+\text{m}=4(4-\text{m})\Rightarrow \text{m}=\dfrac{12}{5}=2.4\text{kg}$
Mass of the second ball is 2.4 kg.

Hence, the correct answer is option C.

Note: We should know that a collision is the event in which two or more bodies exert forces on each other in about a relatively short time. Although the most common use of the word collision refers to incidents in which two or more objects collide with great force, the scientific use of the term implies nothing about the magnitude of the force. Consider the situation where two bodies collide with each other. During the collision, each body exerts a force on the other. This force is called an impulsive force, because it acts for a short period of time compared to the whole motion of the objects, and its value is usually large. A perfectly inelastic collision happens when the maximum amount of kinetic energy in a system is lost. In such a collision, the colliding particles stick together. The kinetic energy is used on the bonding energy of the two bodies.