A ball is thrown from ground at an angle θ with horizontal and with an initial speed u 0 . For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V 1 . After hitting the ground, the ball rebounds at the same angle θ but with a reduced speed of u 0 / α . Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is 0.8 V 1 , the value of α is \ \ \ \ —.

Physics · Motion in Plane

JEE Advanced 2019 — Paper 2 — Question 10

A ball is thrown from ground at an angle $\theta$ with horizontal and with an initial speed $\mathrm{u}_{0}$ . For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is $\mathrm{V}_{1}$ . After hitting the ground, the ball rebounds at the same angle $\theta$ but with a reduced speed of $u_{0} / \alpha$ . Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is $0.8 \mathrm{~V}_{1}$ , the value of $\alpha$ is $\_\_\_\_$ —.

Answer: 4

Numerical answer — enter this value.

Step-by-step solution

Let $\frac{2 \mathrm{u}_{0} \sin \theta}{\mathrm{~g}}=\mathrm{T}_{0}$ and $\mathrm{u}_{0} \cos \theta=\mathrm{v}_{1}$ (given)

Average velocity $=\frac{\mathrm{u}_{0} \cos \theta T_{0}+\frac{\mathrm{u}_{0}}{\alpha} \cos \theta \frac{\mathrm{~T}}{\alpha}+\frac{\mathrm{u}_{0}}{\alpha^{2}} \cos \theta \frac{\mathrm{~T}_{0}}{\alpha^{2}}}{T_{0}+\frac{T_{0}}{\alpha}+\frac{T_{0}}{\alpha^{2}}+\ldots \ldots}$

$=\frac{u_{0} \cos _{0} \frac{1}{1+\frac{1}{\alpha^{2}}+\frac{1}{\alpha^{4}}+\ldots \ldots \ldots .}}{\mathrm{T}_{0}-1+\frac{1}{\alpha}+\frac{1}{\alpha^{2}}+\ldots . .}=v_{1} \frac{\alpha}{\alpha+1}=0.8 \mathrm{~V}$

$\alpha=4$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2019
Paper: Paper 2
Subject: Physics
Chapter: Motion in Plane
Topic: Oblique and Horizontal Projectile Motion