Question

Three identical balls numbered as I, II and III are placed on a smooth floor on a straight line at a separation of $10 \mathrm{m}$ between balls as shown in figure. Initially balls are stationary. Ball I is given velocity of $10 \mathrm{m} / \mathrm{s}$ towards ball II, collision between ball I and II is inelastic with coefficient of restitution 0.5 but collision between ball II and III is perfectly elastic. What is the time interval between two consecutive collisions between I and II?
(A) $4 \mathrm{sec}$
(B) $24 \mathrm{sec}$
(C) $14 \mathrm{sec}$
(D) $44 \mathrm{sec}$

Answer 1

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.

Complete step by step answer
We know that momentum is a physics term that refers to the quantity of motion that an object has. A sports team that is on the move has the momentum. If an object is in motion (on the move) then it has momentum. One example is the use of air bags in automobiles. Air bags are used in automobile.s because they are able to minimize the effect of the force on an object involved in a collision. Air bags accomplish this by extending the time required to stop the momentum of the driver and passenger. Momentum is mass in motion, and any moving object can have momentum. An object's change in momentum is equal to its impulse. Impulse is a quantity of force times the time interval. Impulse is not equal to momentum itself; rather, it's the increase or decrease of an object's momentum. Based on this concept we have to solve this question

The diagram for this question is given as:

Let the mass of each ball be $\mathrm{m}$.
Collision of ball I with II
Let the velocity of Ball I be $\mathrm{v}_{1}$ and that of ball II be $\mathrm{v}_{2}$ after collision
The coefficient of restitution $\mathrm{e}=0.5$
Velocity of ball I before collision $\mathrm{u}_{1}=10 \mathrm{m} / \mathrm{s}$
Therefore; $\mathrm{e}=\dfrac{\mathrm{v}_{2}-\mathrm{v}_{1}}{\mathrm{u}_{1}-\mathrm{u}_{2}}$
$0.5=\dfrac{\mathrm{v}_{2}-\mathrm{v}_{1}}{10-0}$
By the conservation of momentum:
$\mathrm{m}(10)=\mathrm{m}\left(\mathrm{v}_{1}\right)+\mathrm{m}\left(\mathrm{v}_{2}\right)$
$\mathrm{v}_{1}+\mathrm{v}_{2}=10$
Therefore,
$\mathrm{v}_{2}=7.5 \mathrm{m} / \mathrm{s}$
$\mathrm{v}_{1}=2.5 \mathrm{m} / \mathrm{s}$
The collision between balls II and III is elastic:
Therefore, after collision, ball II becomes at rest at the distance 20m from ball I.

Thus,
Time taken to reach the ball I to ball II will be $\mathrm{t}=\dfrac{10}{2.5}=4 \mathrm{s}$

Note: We should know that a projectile is an object upon which the only force is gravity. Gravity acts to influence the vertical motion of the projectile, thus causing a vertical acceleration. The horizontal motion of the projectile is the result of the tendency of any object in motion to remain in motion at constant velocity. Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory. A projectile is any object that has been thrown, shot, or launched, and ballistics is the study of projectile motion. Examples of projectiles range from a golf ball in flight, to a curve ball thrown by a baseball pitcher to a rocket fired into space.