A particle moves so that its position vector varies with time as vecr=A cos ω t hati+A sin ω t hatj. The initial velocity of the particle is

Question

A particle moves so that its position vector varies with time as vecr=A cos ω t hati+A sin ω t hatj. The initial velocity of the particle is (A) A ω hati (B) A ω hatj (C) A ω( hati+ hatj) (D) A ω( hati- hatj)

Tardigrade Admin · Accepted Answer

Substituting vecr=(A cos ω t hati+A sin ω t hatj) in vecv=(d vecr/d t) we have vecv =A (d/d t)( cos ω t) hati+A (d/d t)( sin ω t) hatj =-A ω sin ω t hati+A ω cos ω t hatj At t=0, v=-A ω sin 0 hati+A ω cos 0 hatj=A ω hatj

Q. A particle moves so that its position vector varies with time as $r = A cos ω t \hat{i} + A sin ω t \hat{j}$ . The initial velocity of the particle is

$A ω \hat{i}$

$A ω \hat{j}$

$A ω (\hat{i} + \hat{j})$

$A ω (\hat{i} - \hat{j})$

Substituting $r = (A cos ω t \hat{i} + A sin ω t \hat{j})$ in $v = \frac{d r}{d t}$ we have
$v = A \frac{d}{d t} (cos ω t) \hat{i} + A \frac{d}{d t} (sin ω t) \hat{j}$
$= - A ω sin ω t \hat{i} + A ω cos ω t \hat{j}$
At $t = 0, v = - A ω sin 0 \hat{i} + A ω cos 0 \hat{j} = A ω \hat{j}$

Q. A particle moves so that its position vector varies with time as r=Acosωti^+Asinωtj^​. The initial velocity of the particle is

Aωi^

Aωj^​

Aω(i^+j^​)

Aω(i^−j^​)