Chapter 3, Problem 110P

(a)

To determine

The velocity of the particle.

Explanation of Solution

Given:

The position of the particle at $t=0$ is ${\overrightarrow{r}}_1=\left(4.0\text{m}\right)\widehat{i}+\left(3.0\text{m}\right)\widehat{j}$.

The position of the particle at $t=2$ is ${\overrightarrow{r}}_2=\left(10\text{m}\right)\widehat{i}-\left(2\text{m}\right)\widehat{j}$ .

The velocity of the particle is ${\overrightarrow{v}}_2=\left(5.0\text{m}/\text{s}\right)\widehat{i}-\left(6.0\text{m}/\text{s}\right)\widehat{j}$ .

Formula used:

Write the expression for the average velocity.

  ${\overrightarrow{v}}_{av}=\frac{\Delta \overrightarrow{r}}{\Delta t}$

Here, $\Delta \overrightarrow{r}$ is the change in the position vector and $\Delta t$ is the change in time.

  ${\overrightarrow{v}}_{av}=\frac{{\overrightarrow{r}}_2-{\overrightarrow{r}}_1}{t_2-t_1}$   ...... (1)

Here, ${\overrightarrow{r}}_2$ is the final position vector of the particle, ${\overrightarrow{r}}_1$ is the initial position vector of the particle, $t_2$ is the final time of the particle and $t_1$ is the initial time of the particle.

Write the expression for the average velocity of the particle.

  $v_{av}=\frac{v_1+v_2}{2}$

Here, $v_{av}$ is the average velocity, $v_1$ is the initial velocity of the particle and $v_2$ is the final velocity.

Solve the above equation for ${\overrightarrow{v}}_1$ .

  ${\overrightarrow{v}}_1=2{\overrightarrow{v}}_{av}-{\overrightarrow{v}}_2$   ...... (2)

Calculation:

Substitute $\left(10\text{m}\right)\widehat{i}-\left(2\text{m}\right)\widehat{j}$ for ${\overrightarrow{r}}_2$ , $\left(4.0\text{m}\right)\widehat{i}+\left(3.0\text{m}\right)\widehat{j}$ for ${\overrightarrow{r}}_1$ , $0$ for $t_1$ and $2\text{s}$ for $t_2$ in equation (1).

  $\begin{array}{l}{\overrightarrow{v}}_{av}=\frac{\left(10\text{m}\right)\widehat{i}-\left(2\text{m}\right)\widehat{j}-\left(4.0\text{m}\right)\widehat{i}+\left(3.0\text{m}\right)\widehat{j}}{2s}\\ {\overrightarrow{v}}_{av}=\left(3.0\text{m}/\text{s}\right)\widehat{i}-\left(2.5\text{m}/\text{s}\right)\widehat{j}\end{array}$

Substitute $\left(3.0\text{m}/\text{s}\right)\widehat{i}-\left(2.5\text{m}/\text{s}\right)\widehat{j}$ for ${\overrightarrow{v}}_{av}$ and $\left(5.0\text{m}/\text{s}\right)\widehat{i}-\left(6.0\text{m}/\text{s}\right)\widehat{j}$ for ${\overrightarrow{v}}_2$ in equation (2).

  $\begin{array}{l}{\overrightarrow{v}}_1=2\left[\left(3.0\text{m}/\text{s}\right)\widehat{i}-\left(2.5\text{m}/\text{s}\right)\widehat{j}\right]-\left[\left(5.0\text{m}/\text{s}\right)\widehat{i}-\left(6.0\text{m}/\text{s}\right)\widehat{j}\right]\\ {\overrightarrow{v}}_1=\left(1.0\text{m}/\text{s}\right)\widehat{i}+\left(1.0\text{m}/\text{s}\right)\widehat{j}\end{array}$

Conclusion:

Thus, the initial velocity is $\left(1.0\text{m}/\text{s}\right)\widehat{i}+\left(1.0\text{m}/\text{s}\right)\widehat{j}$ .

(b)

To determine

The acceleration of the particle.

Explanation of Solution

Given:

The position of the particle at $t=0$ is ${\overrightarrow{r}}_1=\left(4.0\text{m}\right)\widehat{i}+\left(3.0\text{m}\right)\widehat{j}$.

The position of the particle at is ${\overrightarrow{r}}_2=\left(10\text{m}\right)\widehat{i}-\left(2\text{m}\right)\widehat{j}$ .

The velocity of the particle is ${\overrightarrow{v}}_2=\left(5.0\text{m}/\text{s}\right)\widehat{i}-\left(6.0\text{m}/\text{s}\right)\widehat{j}$ .

Formula used:

Write the expression for the acceleration of the particle.

  $a=\frac{\Delta v}{\Delta t}$

Here, $\Delta \overrightarrow{v}$

is the change in the velocity vector and is the change in time.

  $a=\frac{{\overrightarrow{v}}_2-{\overrightarrow{v}}_1}{t_2-t_1}$   ...... (3)

Here, ${\overrightarrow{v}}_2$ is the final velocity, ${\overrightarrow{v}}_1$ is the initial velocity, $t_2$ is the final time and $t_1$ is the initial time.

Calculation:

Substitute $\left(5.0\text{m}/\text{s}\right)\widehat{i}-\left(6.0\text{m}/\text{s}\right)\widehat{j}$ for ${\overrightarrow{v}}_2$ , $\left(1.0\text{m}/\text{s}\right)\widehat{i}+\left(1.0\text{m}/\text{s}\right)\widehat{j}$ for ${\overrightarrow{v}}_1$ , $2\text{s}$ for $t_2$ and $0$ for $t_1$ in equation (3).

  $\begin{array}{l}a=\frac{\left(5.0\text{m}/\text{s}\right)\widehat{i}-\left(6.0\text{m}/\text{s}\right)\widehat{j}-\left(\left(1.0\text{m}/\text{s}\right)\widehat{i}+\left(1.0\text{m}/\text{s}\right)\right)\widehat{j}}{2\text{s}}\\ a=\left(\text{2}\text{.0}\text{m}/{\text{s}}^{\text{2}}\right)\widehat{i}-\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right)\widehat{j}\end{array}$

Conclusion:

The acceleration of the particle is $\left(\text{2}\text{.0}\text{m}/{\text{s}}^{\text{2}}\right)\widehat{i}-\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right)\widehat{j}$ .

(c)

To determine

The velocity of the particle as the function of time.

Explanation of Solution

Given:

The position of the particle at is.

The position of the particle at is

The velocity of the particle is ${\overrightarrow{v}}_2=\left(5.0\text{m}/\text{s}\right)\widehat{i}-\left(6.0\text{m}/\text{s}\right)\widehat{j}$ .

Formula used:

Write the expression for the velocity of the particle as the function of time.

  $\overrightarrow{v}\left(t\right)={\overrightarrow{v}}_1+at$   ...... (4)

Here, $\overrightarrow{v}\left(t\right)$ is the velocity of the particle as the function of time.

Calculation:

Substitute $\left(1.0\text{m}/\text{s}\right)\widehat{i}+\left(1.0\text{m}/\text{s}\right)\widehat{j}$ for ${\overrightarrow{v}}_1$ and $\left(\text{2}\text{.0}\text{m}/{\text{s}}^{\text{2}}\right)\widehat{i}-\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right)\widehat{j}$ for $a$ in equation (4).

  $\begin{array}{c}\overrightarrow{v}\left(t\right)=\left(1.0\text{m}/\text{s}\right)\widehat{i}+\left(1.0\text{m}/\text{s}\right)\widehat{j}+\left(\left(\text{2}\text{.0}\text{m}/{\text{s}}^{\text{2}}\right)\widehat{i}-\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right)\widehat{j}\right)t\\ \overrightarrow{v}\left(t\right)\text{=}\left(\text{1}\text{.0}\text{m}/\text{s}\text{+}\left(\text{2}\text{.0}\text{m}/{\text{s}}^{\text{2}}\right)t\right)\widehat{i}+\left(\text{1}\text{.0}\text{m}/\text{s}-\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right)t\right)\widehat{j}\end{array}$

Conclusion:

Thus, the velocity of the particle as the function of timeis $\left(\text{1}\text{.0}\text{m}/\text{s}\text{+}\left(\text{2}\text{.0}\text{m}/{\text{s}}^{\text{2}}\right)t\right)\widehat{i}+\left(\text{1}\text{.0}\text{m}/\text{s}-\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right)t\right)\widehat{j}$ .

(d)

To determine

The position vector of the particle as the function of time.

Explanation of Solution

Given:

The position of the particle at is.

The position of the particle at is

The velocity of the particle is ${\overrightarrow{v}}_2=\left(5.0\text{m}/\text{s}\right)\widehat{i}-\left(6.0\text{m}/\text{s}\right)\widehat{j}$ .

Formula used:

Write the expression for the position vector as the function of time.

  ${\overrightarrow{r}}_1\left(t\right)={\overrightarrow{r}}_1+{\overrightarrow{v}}_1\left(t\right)+\frac{1}{2}a{t}^2$   ...... (5)

Calculation:

Substitute $\left(4.0\text{m}\right)\widehat{i}+\left(3.0\text{m}\right)\widehat{j}$ for ${\overrightarrow{r}}_1$ , $\left(\text{1}\text{.0}\text{m}/\text{s}\text{+}\left(\text{2}\text{.0}\text{m}/{\text{s}}^{\text{2}}\right)t\right)\text{<mover accent="true"><mo>i</mo><mo>^</mo></mover>+}\left(\text{1}\text{.0}\text{m}/\text{s}\text{-}\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right)\text{t}\right)\text{<mover accent="true"><mo>j</mo><mo>^</mo></mover>}$ for ${\overrightarrow{v}}_1\left(t\right)$ and $\left(2.0\text{m}/{\text{s}}^{\text{2}}\right)\widehat{i}-\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right)\widehat{j}$ for $a$ in equation (5).

  $\begin{array}{l}{\overrightarrow{r}}_1\left(t\right)=\left(\begin{array}{l}\left(\left(4.0\text{m}\right)+\left(\text{1}\text{.0}\text{m}/\text{s}\right)t\text{+}\left(\text{2}\text{.0}\text{m}/{\text{s}}^{\text{2}}\right)t^2+\frac{1}{2}\left(2.0\text{m}/{\text{s}}^{\text{2}}\right)t^2\right)\widehat{i}\\ \text{+}\left(\left(3.0\text{m}+\left(\text{1}\text{.0}\text{m}/\text{s}\right)t-\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right){\text{t}}^2\right)-\left(\text{3}\text{.5}\text{m}/{\text{s}}^{\text{2}}\right)t^2\right)\widehat{j}\end{array}\right)\\ {\overrightarrow{r}}_1\left(t\right)=\left(\left(4.0\text{m}\right)+\left(\text{1}\text{.0}\text{m}/\text{s}\right)t+\left(1.0\text{m}/{\text{s}}^{\text{2}}\right)t^2\right)\widehat{i}+\left(3.0\text{m}+\left(\text{1}\text{.0}\text{m}/\text{s}\right)t-\left(1.8\text{m}/{\text{s}}^{\text{2}}\right)t^2\right)\widehat{j}\end{array}$

Conclusion:

The position vector of the particle as the function of time is:

  $\left(\left(4.0\text{m}\right)+\left(\text{1}\text{.0}\text{m}/\text{s}\right)t+\left(1.0\text{m}/{\text{s}}^{\text{2}}\right)t^2\right)\widehat{i}+\left(3.0\text{m}+\left(\text{1}\text{.0}\text{m}/\text{s}\right)t-\left(1.8\text{m}/{\text{s}}^{\text{2}}\right)t^2\right)\widehat{j}$