JEE Advanced 2022 · Paper 1 · Physics Q16

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PhysicsList match+3 / -1Question 34 of 54

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $\mathrm{m\,s^{-1}}$ ) at time $t = \dfrac\pi3$ s.

Which one of the following options is correct?

List-I

I
$A$ and $B$ are moving on a horizontal circle of radius 1 m with uniform angular speed $\omega = 1\ \mathrm{rad\,s^{-1}}$ . The initial angular positions of $A$ and $B$ at time $t = 0$ are $\theta = 0$ and $\theta = \frac\pi2$ , respectively.
II
Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t = 0$ and $t = 0.1$ s respectively, with the same speed $v = \frac{5\pi}{\sqrt2}\ \mathrm{m\,s^{-1}}$ and at $45^\circ$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide. ( $g = 10\ \mathrm{m\,s^{-2}}$ ).
III
Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_A = x_0\sin\frac{t}{t_0}$ and $x_B = x_0\sin\left(\frac{t}{t_0} + \frac\pi2\right)$ respectively, starting from $t = 0$ . Take $x_0 = 1$ m, $t_0 = 1$ s.
IV
Particle $A$ is rotating in a horizontal circular path of radius 1 m on the $xy$ plane, with constant angular speed $\omega = 1\ \mathrm{rad\,s^{-1}}$ . Particle $B$ is moving up at a constant speed $3\ \mathrm{m\,s^{-1}}$ in the vertical direction as shown in the figure. (Ignore gravity.)

List-II

P
$\frac{\sqrt3+1}{2}$
Q
$\frac{\sqrt3-1}{\sqrt2}$
R
$\sqrt{10}$
S
$\sqrt2$
T
$\sqrt{25\pi^2+1}$

I → R, II → T, III → P, IV → S

I → S, II → P, III → Q, IV → R

I → S, II → T, III → P, IV → R

I → T, II → P, III → R, IV → S

Answer and worked solution

Official answer: C

(I) Both particles move at speed 1 m/s on the circle, a quarter-turn apart, so their velocities are always perpendicular:
$|\vec v_{rel}|=\sqrt2\ \to\ \textbf{S}.$

(II) A is fired to the right and B to the left, each at $45^\circ$ with speed $\frac{5\pi}{\sqrt2}$ , so each has horizontal speed $\frac{5\pi}2$ .

Relative horizontal speed: $5\pi$ .
Relative vertical speed: B started 0.1 s later, so $g\times0.1=1$ m/s.

At $t=\frac\pi3$ both are still in flight (A lands at $\frac\pi2$ s):
$|\vec v_{rel}|=\sqrt{25\pi^2+1}\ \to\ \textbf{T}.$

(III) $v_A=\cos t$ and $v_B=-\sin t$ , so the relative speed is $\cos t+\sin t=\frac{1+\sqrt3}2$ at $\frac\pi3$ → P.

(IV) A moves horizontally at 1 m/s and B vertically at 3 m/s:
$|\vec v_{rel}|=\sqrt{1+9}=\sqrt{10}\ \to\ \textbf{R}.$

Topics:Relative velocity Uniform circular motion Simple harmonic motion

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