Chapter 2, Problem 18E

(a)

To determine

The velocity of the car after each second.

Answer to Problem 18E

The velocities after 1s, 2s, 3s, 4s and 5s are $3.0\text{m/s}$, $6.0\text{m/s}$, $9.0\text{m/s}$, $12\text{m/s}$ and $15\text{m/s}$ respectively.

Explanation of Solution

Given info: The car accelerates at $3.0{\text{m/s}}^{\text{2}}$ and the time is 5 seconds.

Write the formula for average acceleration.

$a=\frac{v_f-v_i}{t}$

Here,

a is the average acceleration

$v_f$ is the final velocity

$v_i$ is the initial velocity

t is the time

Re-arrange the equation to get $v_f$.

$v_f=v_i+at$

For $t=1\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and 1 s for t to get $v_f$.

$\begin{array}{c}{v}_f=\left(0\text{m/s}\right)+\left(3.0{\text{m/s}}^{\text{2}}\right)\left(1\text{s}\right)\\ =3.0\text{m/s}\end{array}$

For $t=2\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and $2\text{s}$ for t to get $v_f$.

$\begin{array}{c}{v}_f=\left(0\text{m/s}\right)+\left(3.0{\text{m/s}}^{\text{2}}\right)\left(2\text{s}\right)\\ =6.0\text{m/s}\end{array}$

For $t=3\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and $3\text{s}$ for t to get $v_f$.

$\begin{array}{c}{v}_f=\left(0\text{m/s}\right)+\left(3.0{\text{m/s}}^{\text{2}}\right)\left(3\text{s}\right)\\ =9.0\text{m/s}\end{array}$

For $t=4\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and $4\text{s}$ for t to get $v_f$.

$\begin{array}{c}{v}_f=\left(0\text{m/s}\right)+\left(3.0{\text{m/s}}^{\text{2}}\right)\left(4\text{s}\right)\\ =12\text{m/s}\end{array}$

For $t=5\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and $5\text{s}$ for t to get $v_f$.

$\begin{array}{c}{v}_f=\left(0\text{m/s}\right)+\left(3.0{\text{m/s}}^{\text{2}}\right)\left(5\text{s}\right)\\ =15\text{m/s}\end{array}$

The velocity time graph is given below.

Figure (1)

Conclusion:

Therefore, the velocities after 1s, 2s, 3s, 4s and 5s are $3.0\text{m/s}$, $6.0\text{m/s}$, $9.0\text{m/s}$, $12\text{m/s}$ and $15\text{m/s}$ respectively.

(b)

To determine

The distance travelled after each second.

Answer to Problem 18E

The distances after 1s, 2s, 3s, 4s and 5s are $1.5\text{m}$, $6.0\text{m}$, $13.5\text{m}$, $24\text{m}$ and $37.5\text{m}$ respectively.

Explanation of Solution

Given info: Acceleration is $4{\text{m/s}}^{\text{2}}$ and time is 5 seconds.

Write the formula to calculate the distance.

$d=v_{0}t+\frac{1}{2}a{t}^2$

Here,

d is the distance

For $t=1\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and $1\text{s}$ for t to get $d$.

$\begin{array}{c}d=\left(0\text{m/s}\right)\left(1\text{s}\right)+\frac{1}{2}\left(3.0{\text{m/s}}^{\text{2}}\right){\left(1\text{s}\right)}^2\\ =1.5\text{m}\end{array}$

For $t=2\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and $2\text{s}$ for t to get $d$.

$\begin{array}{c}d=\left(0\text{m/s}\right)\left(2\text{s}\right)+\frac{1}{2}\left(3.0{\text{m/s}}^{\text{2}}\right){\left(2\text{s}\right)}^2\\ =6.0\text{m}\end{array}$

For $t=3\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and $3\text{s}$ for t to get $d$.

$\begin{array}{c}d=\left(0\text{m/s}\right)\left(3\text{s}\right)+\frac{1}{2}\left(3.0{\text{m/s}}^{\text{2}}\right){\left(3\text{s}\right)}^2\\ =13.5\text{m}\end{array}$

For $t=4\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and $4\text{s}$ for t to get $d$.

$\begin{array}{c}d=\left(0\text{m/s}\right)\left(4\text{s}\right)+\frac{1}{2}\left(3.0{\text{m/s}}^{\text{2}}\right){\left(4\text{s}\right)}^2\\ =24\text{m}\end{array}$

For $t=5\text{s}$,

Substitute 0 m/s for $v_i$, $3.0{\text{m/s}}^{\text{2}}$ for a and $5\text{s}$ for t to get $d$.

$\begin{array}{c}d=\left(0\text{m/s}\right)\left(5\text{s}\right)+\frac{1}{2}\left(3.0{\text{m/s}}^{\text{2}}\right){\left(5\text{s}\right)}^2\\ =37.5\text{m}\end{array}$

The distance time graph is,

Figure(2)

Conclusion:

Therefore, the distances after 1s, 2s, 3s, 4s and 5s are $1.5\text{m}$, $6.0\text{m}$, $13.5\text{m}$, $24\text{m}$ and $37.5\text{m}$ respectively.