Question

The equation of motion of a particle p(x,y) on
a plane are given by
x=4+b cost, y = 5+ b sint. Its velocity at
time 't' is​

Answer 1

The equation of motion of a particle p(x,y) on a plane are given by

x=4+ bcost, y = 5+bsint. Its velocity at time 't' is b

Therefore, option (3) is correct.

Step-by-step explanation:

The equation of motion of the particle can be written as

$\vec r=x\hat i+y\hat j$

or, $\vec r=(4+b\cos t)\hat i+(5+b\sin t)\hat j$

We know that velocity is rate of change of displacement

Therefore,

$\vec v=\frac{d\vec r}{dt}$

$\implies \vec v=\frac{d}{dt}[(4+b\cos t)\hat i+(5+b\sin t)\hat j]$

$\implies \vec v=[\frac{d}{dt}(4+b\cos t)]\hat i+[\frac{d}{dt}(5+b\sin t)]\hat j$

$\implies \vec v=b(-\sin t)\hat i+(b\cos t)\hat j$

$\implies \vec v=-b\sin t \hat i+b\cos t \hat j$

Therefore, the magnitude of the velocity

$|\vec v|=\sqrt{(-b\sin t)^2+(b\cos t)^2}$

$\implies |\vec v|=\sqrt{b^2(\sin^2 t+\cos^2 t)}$

$\implies |\vec v|=b$

Therefore velocity at time t is b

Hope this helps.

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Answer 2

Given: $x=4+b\ cost,y=5+b\ sint$ and $p(x,y)$

To Find: Find the velocity at time $'t'$.

Step-by-step explanation:

We know,

The equation of motion of a moving particle is,

$\vec r=x\hat i+y\hat j$

Plug the value of $x$ and $y$,

⇒$\vec r=(4+b\ cost)\hat i+(5+b\ sint)\hat j$

Velocity of a moving particle is given by rate of change of displacement.

So,

   $\vec v=\frac{d\vec r}{dt}$

Plug the $\vec r$ value in above equation,

⇒$\vec v=\frac{d}{dt}[(4+b\cos t)\hat i+(5+b\sin t)\hat j]$

⇒$\vec v=[\frac{d}{dt}(4+b\cos t)]\hat i+[\frac{d}{dt}(5+b\sin t)]\hat j$

⇒$\vec v=b(-\sin t)\hat i+(b\cos t)\hat j$

⇒$\vec v=-b\sin t \hat i+b\cos t \hat j$

Therefore, the magnitude of the velocity is given,

⇒$|\vec v|=\sqrt{(-b\sin t)^2+(b\cos t)^2}$

⇒$|\vec v|=\sqrt{b^2(\sin^2 t+\cos^2 t)}$

⇒$|\vec v|=b$

Thus, The velocity at time $'t'$ is $b$ unit/sec.