Question

To find the velocity of a particle at\(t = 2\).

Short answer

The velocity of a particle at \(t = 2\) is \(\langle - 2,1\rangle \).

Step-by-step solution

Given equations

Equations are \(r(t) = \left\langle { - \frac{1}{2}{t^2},t} \right\rangle \) and \(t = 2\).

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

Substitute\(\left\langle { - \frac{1}{2}{t^2},t} \right\rangle \)for\({\rm{r}}(t)\)in equation (1).

\(\begin{array}{c}{\rm{v}}(t) = \frac{d}{{dt}}\left( {\left\langle { - \frac{1}{2}{t^2},t} \right\rangle } \right)\\ = \left\langle {\frac{d}{{dt}}\left( { - \frac{1}{2}{t^2}} \right),\frac{d}{{dt}}(t)} \right\rangle \\ = \left\langle { - \frac{1}{2}(2t),1} \right\rangle \\ = \langle - t,1\rangle \end{array}\)

Substitute 2 for\(t\).

\(v(2) = \langle - 2,1\rangle \)

Thus, the velocity of a particle at \(t = 2\) is \(\langle - 2,1\rangle \).