# A particle of mass 1 kg is moving in the xy – plane whose position at an instant t is r = 3 sin 2t i + 3(1 – cos2t)j , where r is in metre and t is in second. The kinetic energy of the particle at instant t is

Q: A particle of mass 1 kg is moving in the xy – plane whose position at an instant t is $\vec{r} = 3 sin 2t \hat{i} + 3(1 – cos2t)\hat{j}$, where r is in metre and t is in second. The kinetic energy of the particle at instant t is

(a) 4.5 J

(b) 10 J

(c) 18 J

(d) Does not depend on the time

Ans: (c) , (d)

Solution: $\vec{r} = 3 sin 2t \hat{i} + 3(1 – cos2t)\hat{j}$

On differentiating w.r.t time

$\frac{d\vec{r}}{dt} = 6 cos 2t \hat{i} + 6 sin2t \hat{j}$

$\displaystyle \vec{v} = 6 cos 2t \hat{i} + 6 sin2t \hat{j}$

$\displaystyle v = \sqrt{(6 cos 2t)^2 + (6 sin2t)^2} = 6$

Kinetic Energy , $\displaystyle K = \frac{1}{2} m v^2$

$\displaystyle K = \frac{1}{2} \times 1 \times (6)^2$

K = 18 J

It is Independent of time .