Question

You travel from point A to point B in a car moving at a constant speed of $70\mathrm{km}/\mathrm{h}$. Then you travel the same distance from point B to another point C, moving at a constant speed of $90\mathrm{km}/\mathrm{h}$. Is your average speed for the entire trip from A to C, equal to $80\mathrm{km}/\mathrm{h}$? Explain why or why not.

Short answer

The average speed for the entire trip from point A to C is less than $80\mathrm{km}/\mathrm{h}$.

Step-by-step solution

Step 1. Definition of average speed

Speed is a scalar quantity, which means it has only magnitude. Average speed is the ratio of the total distance traveled by an object to the total time taken to cover the distance.

Step 2. Comparison between the times taken for traveling from A to B and B to C

Speed at which your car travels from point A to B,$s_1=70\mathrm{km}/\mathrm{h}$

Speed at which your car covers the same distance from point B to C,$s_2=90\mathrm{km}/\mathrm{h}$

Thus, you spend more time traveling at $70\mathrm{km}/\mathrm{h}$than at $90\mathrm{km}/\mathrm{h}$, for the same distance.

Step 3. Determination of the average speed of the car

Let the distance from point A to B (or point B to C) be y.

Then the total distance travelled will be $2y$.

The time taken (t) by an object to a certain distance is:

$t=\frac{\mathrm{Distance}}{\mathrm{Speed}}$

Time elapsed in traveling from A to B,$t_1=\frac{y}{70\mathrm{km}/\mathrm{h}}$

Similarly, time elapsed in traveling from A to B, $t_2=\frac{y}{90\mathrm{km}/\mathrm{h}}$

The average speed is:$s_{\mathrm{av}}=\frac{\mathrm{Total}\mathrm{distance}}{\mathrm{Total}\mathrm{time}}$

Therefore, the average speed for the entire trip from A to C is:

$\begin{array}{c}=\frac{2y}{\left(\frac{y}{70\mathrm{km}/\mathrm{h}}+\frac{y}{90\mathrm{km}/\mathrm{h}}\right)}\\ =\frac{2\times 90\times 70}{90+70}\mathrm{km}/\mathrm{h}\\ =78.75\mathrm{km}/\mathrm{h}\end{array}$

Thus, the average speed for the entire journey is less than $80\mathrm{km}/\mathrm{h}$.