Question

A particle is moving in a plane with velocity given by $\overrightarrow{u}$= u₀ $\widehat{i}$ + (aωcosωt)$\widehat{j}$, where $\widehat{i}$and $\widehat{j}$are unit vectors along x and y axis, respectively. If the particle is at origin at t = 0, the distance from origin at time 3π/2 ωis

• A. a² + ω²
• B. $\left\lbrack \left( 3\pi u_{0}/2\omega \right)^{2} + a^{2} \right\rbrack^{1/2}$
• C. $\sqrt{a^{2} + \left( 2/3\pi u_{0} \right)^{2}}$
• D. $\sqrt{a^{2} + \left( \pi u_{0}/\omega \right)^{2}}$

Answer 1

Answer: (B)

$\left\lbrack \left( 3\pi u_{0}/2\omega \right)^{2} + a^{2} \right\rbrack^{1/2}$

Answer 2

Given, u = u₀$\widehat{i}$ + (aω cos ωt)$\widehat{j}$

Thus velocity along y axis, U_y = a cos ωt and velocity along x axis, vx = u₀.

Displacement at time t in horizontal direction,

x = $\int_{}^{}{u_{0}dt} = u_{0}t\left( Qv = \frac{dx}{dt} \right)$

and y = $\int_{}^{}{a\omega\cos\omega tdt} = a\sin\omega t$

Eliminating t, y = a sin (ωx/u₀)

At time 3π/2ω, x = u₀(3π/2ω)

and y = a sin 3π/2 = -a

Thus distance of particle from origin

S = $\sqrt{\left\lbrack \frac{3\pi\mu_{0}}{2\omega} \right\rbrack^{2} + a^{2}}\left( QR = \sqrt{x^{2} + y^{2}} \right)$