Chapter 2, Problem 37P

(b)

To determine

The time the rabbit spends running at $\text{0}\text{.60m/sec}$

Answer to Problem 37P

Rabbit spends $\text{1200 s}$ to run at a speed of $\text{0}\text{.60m/sec}$.

Explanation of Solution

Write the expression for distance traveled at each time interval, in terms of velocity and time, and the $v_2$ is zero because the rabbit is at rest.

  $\begin{array}{c}{x}_1=v_1\Delta t_1\\ x_2=v_2\Delta t_2=0\\ x_3=v_3\Delta t_3\end{array}$        (I)

Write the expression for total distance,

  $\begin{array}{c}x=x_1 +x_2 +x_3 \\ =x_1 +0+x_3\end{array}$        (II)

Substitute expression (II) in expression (I),

  $x=v_1\Delta t_1+v_3\Delta t_3$        (III)

Write the expression for average velocity

  $\begin{array}{l}{v}_{\text{ave}}=\frac{\Delta x}{\Delta t}\\ v_{\text{ave}}=\frac{x_f-x_i}{t_f-t_i}\end{array}$        (IV)

Here, $\Delta x$ is the distance, $\Delta t$ is the time interval.

In expression (III) the initial position is zero at  $x=0\text{ at }t=0$ so,can be re-written as,

  $v_{\text{ave}}=\frac{x}{t}$        (V)

The expression for the total time, $t$, is the sum of each individual time interval,

  $t=\Delta t_1+\Delta t_2+\Delta t_3$        (VI)

Substitute expression (V) in (VI), and solve for $\Delta t_1$

  $\begin{array}{l}\frac{x}{v_{\text{ave}}}=\Delta t_1+\Delta t_2+\Delta t_3\\ \Delta t_1=\frac{x}{v_{\text{ave}}}-\Delta t_2-\Delta t_3\end{array}$        (VI)

Solve for $\Delta t_3$ by substituting expression (VI) in (III)

  $\begin{array}{c}x=v_1\left(\frac{x}{v_{\text{ave}}}-\Delta t_2-\Delta t_3\right)+v_3\Delta t_3\\ x=v_1\frac{x}{v_{\text{ave}}}-v_1\Delta t_2-v_1\Delta t_3+v_3\Delta t_3\\ v_1\Delta t_3-v_3\Delta t_3=v_1\frac{x}{v_{\text{ave}}}-v_1\Delta t_2-x\\ \Delta t_3=\frac{v_1\frac{x}{v_{\text{ave}}}-v_1\Delta t_2-x}{v_1-v_3}\end{array}$

Conclusion:

Substitute the $1.5\text{ m/s}$ for $v_1$, $\text{1200 m}$ for $x$, $\text{0}\text{.80 m/s}$ for $v_{\text{ave}}$, $\text{10 s}$ for $\Delta t_2$, $0.60\text{m/s}$ for $v_3$ values in the final expression of $\Delta t_3$

  $\begin{array}{c}\Delta t_3=\frac{(1.5\text{ m/s)}\frac{\text{1200 m}}{\text{0}\text{.80 m/s}}-(1.5\text{m/s)(10 s)}-1200\text{m}}{\text{1}\text{.5 m/s}-0.60\text{m/s}}\\ \Delta t_3=1150\text{s}\\ \text{=1200 s}\end{array}$

Rabbit spends $\text{1200 s}$ to run at a speed of $\text{0}\text{.60m/sec}$.

(b)

To determine

The time for the rabbit to run at $\text{0}{\text{.60m/s}}^{\text{2}}$

Answer to Problem 37P

The time for the rabbit to run at $\text{0}{\text{.60m/s}}^{\text{2}}$ is $340\sec$

Explanation of Solution

Write the expression for distance traveled at each time interval, in terms of velocity and time, and the $v_2$ is zero because the rabbit is at rest.

  $\begin{array}{c}{x}_1=v_1\Delta t_1\\ x_2=v_2\Delta t_2=0\\ x_3=v_3\Delta t_3\end{array}$        (I)

Write the expression for total distance,

  $\begin{array}{c}x=x_1 +x_2 +x_3 \\ =x_1 +0+x_3\end{array}$        (II)

Substitute expression (II) in expression (I),

  $x=v_1\Delta t_1+v_3\Delta t_3$        (III)

Write the expression for average velocity

  $\begin{array}{l}{v}_{\text{ave}}=\frac{\Delta x}{\Delta t}\\ v_{\text{ave}}=\frac{x_f-x_i}{t_f-t_i}\end{array}$        (IV)

Here, $\Delta x$ is the distance, $\Delta t$ is the time interval.

In expression (III) the initial position is zero at  $x=0\text{ at }t=0$ so, can be re-written as,

  $v_{\text{ave}}=\frac{x}{t}$        (V)

The expression for the total time, $t$, is the sum of each individual time interval,

  $t=\Delta t_1+\Delta t_2+\Delta t_3$        (VI)

Substitute expression (V) in (VI), and solve for $\Delta t_1$

  $\begin{array}{l}\frac{x}{v_{\text{ave}}}=\Delta t_1+\Delta t_2+\Delta t_3\\ \Delta t_1=\frac{x}{v_{\text{ave}}}-\Delta t_2-\Delta t_3\end{array}$        (VI)

Conclusion:

Substitute $\text{1200 m}$ for $x$, $\text{0}\text{.80 m/s}$ for $v_{\text{ave}}$, $\text{10 s}$ for $\Delta t_2$, $1150\sec$ for $\Delta t_2$ values in the final expression of $\Delta t_3$

  $\begin{array}{c}\frac{x}{v_{\text{ave}}}=\Delta t_1+\Delta t_2+\Delta t_3\\ \Delta t_1=\frac{x}{v_{\text{ave}}}-\Delta t_2-\Delta t_3\\ \Delta t_1=\frac{1200\text{ m}}{0.8\text{ m/s}}-10\text{ s}-1150\text{ s}\\ \Delta t_1=340\text{ s}\end{array}$

The time for the rabbit to run at $\text{0}{\text{.60m/s}}^{\text{2}}$ is $340\sec$