Question

The equation of motion of a particle p(x,y) on
a plane are given by
x=4+ bcost, y = 5+b sint. Its velocity at
time 't' is
1)4
2)5
3)b
4) tant​

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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