Chapter 4, Problem 4.1CYU

To determine

(a)

The instantaneous velocity at t = 3 s. if the position function of a particle is:

$\overrightarrow{r}(t)=3t^3\widehat{i}+4\widehat{j}$

Answer to Problem 4.1CYU

The instantaneous velocity at t = 3 s will be $\left(81\frac{m}{s}\right)\widehat{i}$

Explanation of Solution

Given info:

$\overrightarrow{r}(t)=3t^3\widehat{i}+4\widehat{j}$

Formula used:

The instantaneous velocity is given by

$v(t)=\frac{d\overrightarrow{r}(t)}{dt}$

Calculation:

Substituting the values in above formula, we get

$\begin{array}{l}v(t)=9t^2\widehat{i}+0\\ \text{At,}t=3s\\ $ v(3s)=9{(3)}^2\widehat{i}=\left(81\frac{m}{s}\right)\widehat{i}$\end{array}$

Thus, instantaneous velocity at t= 3 s will be $\left(81\frac{m}{s}\right)\widehat{i}$ for the given position function.

To determine

(b)

If the average velocity between 2s and 4s is equal to the instantaneous velocity at t= 3s

Answer to Problem 4.1CYU

The average velocity between t = 2 s and t = 4 s is $\left(84\frac{m}{s}\right)\widehat{i}$ which is not equal to the instantaneous velocity at t = 3 s.

Explanation of Solution

Given info:

$\overrightarrow{r}(t)=3t^3\widehat{i}+4\widehat{j}$

Formula used:

The average velocity is given by:

$v_{\text{ave}}=\frac{r(4)-r(2)}{t(4)-t(2)}$

Calculation:

$\begin{array}{l}\overrightarrow{r}(t)=3t^3\widehat{i}+4\widehat{j}\\ $ \overrightarrow{r}(2s)=3{(2)}^3\widehat{i}+4\widehat{j}=24\widehat{i}+4\widehat{j}$$ \overrightarrow{r}(4s)=3{(4)}^3\widehat{i}+4\widehat{j}=192\widehat{i}+4\widehat{j}$\end{array}$

Then the average velocity is:

$\begin{array}{l}\overrightarrow{r}(t)=3t^3\widehat{i}+4\widehat{j}\\ $ \overrightarrow{r}(2s)=3{(2)}^3\widehat{i}+4\widehat{j}=24\widehat{i}+4\widehat{j}$$ \overrightarrow{r}(4s)=3{(4)}^3\widehat{i}+4\widehat{j}=192\widehat{i}+4\widehat{j}$\overline{v}=\frac{\overrightarrow{r}(4s)-\overrightarrow{r}(2s)}{t(4s)-t(2s)}=\frac{(192\widehat{i}+4\widehat{j})m-(24\widehat{i}+4\widehat{j})m}{4s-2s}\\ \overline{v}=\frac{(168m)\widehat{i}}{2s}=\left(84\frac{m}{s}\right)\widehat{i}\end{array}$

Thus, the average velocity between t = 2 s and t = 4 s is $\left(84\frac{m}{s}\right)\widehat{i}$ which is not equal to the instantaneous velocity at t = 3 s.