A particle moves such that its position vector vecr(t)=cosω t hati + sin ω t hatj where ω is a constant and t is time. Then which of the following statements is true for the velocity vecv(t) and acceleration veca(t) of the particle :

Question

A particle moves such that its position vector vecr(t)=cosω t hati + sin ω t hatj where ω is a constant and t is time. Then which of the following statements is true for the velocity vecv(t) and acceleration veca(t) of the particle : (A)  vecv and veca both are parallel to vecr (B)  vecv is perpendicular to vecr and veca is directed away from the origin (C)  vecv and veca both are perpendicular to vecr (D)  vecv is perpendicular to vecr and veca is directed towards the origin

Tardigrade Admin · Accepted Answer

vecr(t)=cos ω hati+sin ω t hatj On diff. we get vecv=-ω sin ω t hati+ω cos ω t hatj veca=-ω2 vecr vecv. vecr=0

Q. A particle moves such that its position vector $r (t) = cos ω t \hat{i} + s in ω t \hat{j}$ where $ω$ is a constant and $t$ is time. Then which of the following statements is true for the velocity $v (t)$ and acceleration $a (t)$ of the particle :

$v$ and $a$ both are parallel to $r$

$v$ is perpendicular to $r$ and $a$ is directed away from the origin

$v$ and $a$ both are perpendicular to $r$

$v$ is perpendicular to $r$ and $a$ is directed towards the origin

$r (t) = cos ω \hat{i} + s in ω t \hat{j}$
On diff. we get
$v = - ω s in ω t \hat{i} + ω cos ω t \hat{j}$
$a = - ω^{2} r$
$v . r = 0$

Q. A particle moves such that its position vector r(t)=cosωti^+sinωtj^​ where ω is a constant and t is time. Then which of the following statements is true for the velocity v(t) and acceleration a(t) of the particle :

v and a both are parallel to r

v is perpendicular to r and a is directed away from the origin

v and a both are perpendicular to r

v is perpendicular to r and a is directed towards the origin