Question

To find: The velocity of a particle at\(t = \frac{\pi }{3}\).

Short answer

The velocity of a particle at \(t = \frac{\pi }{3}\) is \( - \frac{{3\sqrt 3 }}{2}{\rm{i}} + {\rm{j}}\).

Step-by-step solution

Given data

The given data is \({\rm{r}}(t) = 3\cos t{\rm{i}} + 2\sin \) and \(t = \frac{\pi }{3}\).

Concept of Velocity

The expression to find the velocity:

\({\rm{V}}(t) = {{\rm{r}}^\prime }(t)\)

\({\rm{V}}(t) = \frac{d}{{dt}}({\rm{r}}(t))\) ...... (1)

Substitute the values

Write the expression to find the velocity.

\(\begin{aligned}{c}{\rm{v}}(t) = {{\rm{r}}^\prime }(t)\\{\rm{v}}(t) = \frac{d}{{dt}}({\rm{r}}(t))\end{aligned}\)

Substitute\(3\cos t{\rm{i}} + 2\sin t{\rm{j}} \to {\rm{r}}(t)\)in equation (1).

\(\begin{aligned}{c}{\rm{v}}(t) = \frac{d}{{dt}}(3\cos t{\rm{i}} + 2\sin t{\rm{j}})\\ = \frac{d}{{dt}}(3\cos t{\rm{i}}) + \frac{d}{{dt}}(2\sin t{\rm{j}})\\ = - 3\sin t{\rm{i}} + 2\cos t{\rm{j}}\end{aligned}\)

Substitute,\(\frac{\pi }{3} \to t\).

\(\begin{aligned}{c}v\left( {\frac{\pi }{3}} \right) = - 3\sin \left( {\frac{\pi }{3}} \right){\rm{i}} + 2\cos \left( {\frac{\pi }{3}} \right){\rm{j}}\\ = - 3\left( {\frac{{\sqrt 3 }}{2}} \right){\rm{i}} + 2\left( {\frac{1}{2}} \right){\rm{j}}\\ = \frac{{ - 3\sqrt 3 }}{2}{\rm{i}} + {\rm{j}}\end{aligned}\)

Thus, the velocity of a particle at \(t = \frac{\pi }{3}\) is \( - \frac{{3\sqrt 3 }}{2}{\rm{i}} + {\rm{j}}\).