List I describes four systems, each with two particles A and B in relative motion as shown in figure. List II gives possible magnitudes of their relative velocities (in m s - 1 ) at time t = π 3 s . List-I List-II (I) A and B are moving on a horizontal circle of radius 1 m with uniform angular speed ω = 1 rad s - 1 . The initial angular positions of A and B at time t = 0 are θ = 0 and θ = π 2 respectively. (P) 3 + 1 2 (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 = 5 π 2 m s - 1 and at 45 ° from the horizontal plane. The initial separation between A and B is large enough so that they do not collide, g = 10 m s - 2 . (Q) 3 - 1 2 (III) Two harmonic oscillators A and B moving in the x direction according to x A = x 0 sin t t 0 and x B = x 0 sin t t 0 + π 2 respectively, starting from t = 0 . Take x 0 = 1 m , t 0 = 1 s . (R) 10 (IV) Particle A is rotating in a horizontal circular path of radius 1 m on the x y plane, with constant angular speed ω = 1 rad s - 1 . Particle B is moving up at a constant speed 3 m s - 1 in the vertical direction as shown in the figure. (Ignore gravity.) (S) 2 (T) 25 π 2 + 1 Which one of the following options is correct?

Question

List I describes four systems, each with two particles A and B in relative motion as shown in figure. List II gives possible magnitudes of their relative velocities (in m s - 1 ) at time t = π 3 s . List-I List-II (I) A and B are moving on a horizontal circle of radius 1 m with uniform angular speed ω = 1 rad s - 1 . The initial angular positions of A and B at time t = 0 are θ = 0 and θ = π 2 respectively. (P) 3 + 1 2 (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 = 5 π 2 m s - 1 and at 45 ° from the horizontal plane. The initial separation between A and B is large enough so that they do not collide, g = 10 m s - 2 . (Q) 3 - 1 2 (III) Two harmonic oscillators A and B moving in the x direction according to x A = x 0 sin t t 0 and x B = x 0 sin t t 0 + π 2 respectively, starting from t = 0 . Take x 0 = 1 m , t 0 = 1 s . (R) 10 (IV) Particle A is rotating in a horizontal circular path of radius 1 m on the x y plane, with constant angular speed ω = 1 rad s - 1 . Particle B is moving up at a constant speed 3 m s - 1 in the vertical direction as shown in the figure. (Ignore gravity.) (S) 2 (T) 25 π 2 + 1 Which one of the following options is correct?, 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

MARKS App · Accepted Answer

(I) Angular velocity of both the particles is same. Therefore, angle between them will remain 90 ° . v r e l = 2 ω R = 2 m s - 1 Hence, I → S (II) The velocity of A at any time t can be written as v A t = 5 π 2 cos 45 ° i ^ + 5 π 2 × sin 45 ° - g × t j ^ = 5 π 2 i ^ + 5 π 2 - g t j ^ Velocity of B at any time t v B t = - 5 π 2 i ^ + 5 π 2 - g t - 0 . 1 j ^ Relative velocity at time t will be, v → r e l = 5 π i ^ - g × 0 . 1 j ^ v → r e l = 25 π 2 + 1 m s - 1 Hence, II → T (III) Relative position of A wrt B can be written as x = x A - x B = x 0 sin t - x 0 sin t + π 2 Using trigonometric relation x = 2 x 0 sin t - π 4 Therefore, v rel = d x d t = 2 x 0 cos t - π 4 at t = π 3 v r e l = 2 cos π 3 - π 4 = 2 × 3 + 1 2 2 = 3 + 1 2 m s - 1 Hence, III → P (IV) The velocity of A is in x y plane and it is v A = ω r = 1 m s - 1 and the velocity of B is along z axis and v B = 3 m s - 1 . Therefore, v rel = 3 2 + 1 2 = 10 m s - 1 Hence, IV → R

	List-I		List-II
(I)	$A$ and $B$ are moving on a horizontal circle of radius $1 m$ with uniform angular speed $ω = 1 rad s^{- 1}$ . The initial angular positions of $A$ and $B$ at time $t = 0$ are $θ = 0$ and $θ = \frac{π}{2}$ respectively.	(P)	$\frac{\sqrt{3} + 1}{2}$
(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 $v = \frac{5 π}{\sqrt{2}} m s^{- 1}$ and at $45 °$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide, $(g = 10 m s^{- 2})$ .	(Q)	$\frac{(\sqrt{3} - 1)}{\sqrt{2}}$
(III)	Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_{A} = x_{0} \sin \frac{t}{t_{0}}$ and $x_{B} = x_{0} \sin (\frac{t}{t_{0}} + \frac{π}{2})$ respectively, starting from $t = 0$ . Take $x_{0} = 1 m, t_{0} = 1 s$ .	(R)	$\sqrt{10}$
(IV)	Particle $A$ is rotating in a horizontal circular path of radius $1 m$ on the $x y$ plane, with constant angular speed $ω = 1 rad s^{- 1}$ . Particle $B$ is moving up at a constant speed $3 m s^{- 1}$ in the vertical direction as shown in the figure. (Ignore gravity.)	(S)	$\sqrt{2}$
		(T)	$\sqrt{25 π^{2} + 1}$

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