Chapter 2, Problem 53P

(a)

To determine

The average velocity over the time interval $t=0.0\text{s}$ to $t=10.0\text{s}$.

Answer to Problem 53P

The average velocity over the time interval $t=0.0\text{s}$ to $t=10.0\text{s}$, is $\underset{\_}{3.0\text{m}/\text{s}}$.

Explanation of Solution

Write the expression for average velocity.

  $v_{avg}=\frac{\Delta x}{\Delta t}$        (I)

Conclusion:

Substitute, $30\text{m}-\text{0}\text{m}$ for $\Delta x$, and $10.0\text{s}-0.\text{0}\text{s}$ for $\Delta t$ in equation (I) to find the average velocity from $t=0.0\text{s}$ to $t=10.0\text{s}$.

  $\begin{array}{c}{v}_{avg}=\frac{30\text{m}-\text{0}\text{m}}{10.0\text{s}-\text{0}\text{s}}\\ =3.0\text{m}/\text{s}\end{array}$

Therefore, the average velocity over the time interval $t=0.0\text{s}$ to $t=10.0\text{s}$, is $\underset{\_}{3.0\text{m}/\text{s}}$.

(b)

To determine

The average velocity over the time interval $t=0.0\text{s}$ to $t=5.0\text{s}$.

Answer to Problem 53P

The average velocity over the time interval $t=0.0\text{s}$ to $t=5.0\text{s}$ is $\underset{\_}{4.5\text{m}/\text{s}}$.

Explanation of Solution

Use equation (I) to find the answer.

Conclusion:

Substitute, $22.5\text{m}-\text{0}\text{m}$ for $\Delta x$, and $5.0\text{s}-0.\text{0}\text{s}$ for $\Delta t$ in equation (I) to find the average velocity from $t=0.0\text{s}$ to $t=5.0\text{s}$.

  $\begin{array}{c}{v}_{avg}=\frac{22.5\text{m}-\text{0}\text{m}}{5.0\text{s}-\text{0}\text{s}}\\ =4.5\text{m}/\text{s}\end{array}$

Therefore, the average velocity over the time interval $t=0.0\text{s}$ to $t=5.0\text{s}$ is $\underset{\_}{4.5\text{m}/\text{s}}$.

(c)

To determine

The average velocity between $t=5.0\text{s}$ and $t=10.0\text{s}$.

Answer to Problem 53P

The average velocity between $t=5.0\text{s}$ and $t=10.0\text{s}$ is $\underset{\_}{1.5\text{m}/\text{s}}$.

Explanation of Solution

Use equation (I) to find the answer.

Conclusion:

Substitute, $\text{30}\text{m}-22.5\text{m}$ for $\Delta x$, and $10.0\text{s}-5.\text{0}\text{s}$ for $\Delta t$ in equation (I) to find the average velocity from $t=5.0\text{s}$ to $t=10.0\text{s}$.

  $\begin{array}{c}{v}_{avg}=\frac{\text{30}\text{m}-22.5\text{m}}{10.0\text{s}-5.\text{0}\text{s}}\\ =1.5\text{m}/\text{s}\end{array}$

Therefore, the average velocity between $t=5.0\text{s}$ and $t=10.0\text{s}$ is $\underset{\_}{1.5\text{m}/\text{s}}$.

(d)

To determine

The relation of answers in part (a), (b), and (c).

Answer to Problem 53P

The relation is average of answers in part (b), and (c) is answer in part (a).

Explanation of Solution

The total average velocity over the time interval $t=0.0\text{s}$ to $t=10.0\text{s}$ must be equal to the average of the individual average velocities for the time intervals $t=0.0\text{s}$ to $t=5.0\text{s}$ and $t=5.0\text{s}$ to $t=10.0\text{s}$.

Write the expression for the average velocity.

  $v_{avg,a}=\frac{v_{avg,b}+v_{avg,c}}{2}$        (II)

Conclusion:

Substitute, $4.5\text{m}/\text{s}$ for $v_{avg,b}$, $1.5\text{m}/\text{s}$ for $v_{avg,c}$ in equation (II).

  $\begin{array}{c}{v}_{avg,a}=\frac{4.5\text{m}/\text{s}+1.5\text{m}/\text{s}}{2}\\ =3.0\text{m}/\text{s}\end{array}$

Therefore, the relation is average of answers in part (b), and (c) is answer in part (a).