Question

On a straight road with the +x axis chosen to point in the direction of motion, you drive for 3 h at a constant 30 mi/h, then in a few seconds you speed up to 60mi/h and drive at this speed for 1 h.

(a) What was the x component of average velocity for the 4 h period, using the fundamental definition. Of average velocity, which is the displacement divided by the time interval?

(b) Suppose that instead you use the equation ${\mathit{v}}_{\mathbf{a}\mathbf{v}\mathbf{g}\mathbf{,}\mathbf{x}}\mathbf{=}\frac{\mathbf{(}{\mathbf{v}}_{\mathbf{i}\mathbf{x}}\mathbf{+}{\mathbf{v}}_{\mathbf{f}\mathbf{x}}\mathbf{)}}{\mathbf{2}}$. What do you calculate for the x component of average velocity?

(c) Why does the equation used in part (b) give the wrong answer?

Short answer

1. Average velocity$v_{avg}=37.5\text{\hspace{0.17em}m}/\text{s}$
2. The value of average velocity is $v_{avg,x}=45\text{\hspace{0.17em}\hspace{0.17em}mi}/\text{h}$
3. Because the formula is valid for the constant rate.

Step-by-step solution

Identification of given data

• Initial velocity$v_i=\text{30\hspace{0.17em}mi}/\text{h}$
• Final velocity $v_f=6\text{0\hspace{0.17em}mi}/\text{h}$
• Initial time$t_1=3\text{\hspace{0.17em}h}$
• Final time $t_2=1\text{\hspace{0.17em}h}$

 Average velocity

Average speed is the ratio of distance covered in a particular period of time.

 Calculation for the average velocity

(a)

The average speed of an object is calculated by

$v_{avg}=\frac{d_t}{t}$

Here, $d_t$- Total distance, and t -Total time

Distance travel in first 3h

$\begin{array}{l}{d}_1=30\times 3\\ d_1=90\text{\hspace{0.17em}m}\end{array}$

Distance travel in 1h

$\begin{array}{l}{d}_2=60\times 1\\ d_2=60\text{\hspace{0.17em}mi}\end{array}$

Total distance

$\begin{array}{l}{d}_t=60+90\\ d_t=150\text{\hspace{0.17em}mi}\end{array}$

Now,

$\begin{array}{l}{v}_{avg}=\frac{150}{4}\\ v_{avg}=37.5\text{\hspace{0.17em}mi}/\text{s}\end{array}$

The value of average velocity is$37.5\text{\hspace{0.17em}mi}/\text{s}$

 Calculation for the average velocity

(b)

The given equation from the question

$\begin{array}{l}{v}_{avg,x}=\frac{v_{ix}+v_{fx}}{2}\\ v_{avg,x}=\frac{30+60}{2}\\ v_{avg,x}=45\text{\hspace{0.17em}\hspace{0.17em}mi}/\text{h}\end{array}$

The value of average velocity is$v_{avg,x}=45\text{\hspace{0.17em}\hspace{0.17em}mi}/\text{h}$

 Explanation

(c)

The given formula in the question is being only valid for a constant rate like baseballs, which means the result generated is wrong. It didn’t consider the different time intervals.