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A ball is projected from the floor of a cabin of height 7m with speed $20 \mathrm{m} / \mathrm{s}$ at an angle of $37^{\circ}$ with the floor of the cabin. It makes a successive collision with the wall of the cabin and then returns again to its floor. Assume all collisions are perfectly elastic. Find time taken (in sec) by the ball to reach the floor after collisions with the ceiling of the cabin.
(A) 1
(B) 2
(C) 3
(D) 4

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Last updated date: 21st Jun 2024
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Answer
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Hint 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
To calculate the time taken,
We must consider the following equations as per the given questions
We must calculate the magnitude in x and y axis as
${{\text{U}}_{\text{x}}}=20\cos 37{}^\circ $
$=20\times \dfrac{4}{5}=16m/s$
Hence, we obtain the initial velocity in the x axis as 16m/s.
${{\text{U}}_{\text{y}}}=20\times \sin 37{}^\circ $
$=20\times \dfrac{3}{5}=12m/s$
We obtain the initial velocity in y axis as 12m/s.
Now, we can use the equation of motion of particle,
${{\text{V}}_{\text{y}}}^{2}={{\text{U}}_{\text{y}}}^{2}+\text{2aS}$
${{\text{V}}_{\text{y}}}^{2}={{12}^{2}}-2\times 10\times 7$
${{\text{V}}_{\text{y}}}=2m/s$
We obtain the final velocity in the y-axis as 2m/s.
$\text{S=ut+}\dfrac{1}{2}\text{a}{{\text{t}}^{\text{2}}}$
$\text{t=}\left( \text{2}\times \text{t} \right)-\left( \dfrac{1}{2}\times 10\times {{\text{t}}^{\text{2}}} \right)$
$\text{t=}\left( \text{2t} \right)-\left( \text{5}{{\text{t}}^{\text{2}}} \right)$
$\text{5}{{\text{t}}^{\text{2}}}-\text{2t+t=0}$
$\therefore \text{t=1sec}$

Therefore, the correct answer is Option A.

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.