Solving Speed Problems (Measurement)

I. Calculating an average speed

To calculate the average speed V of a vehicle, when we know the distance d traveled and the duration t of the journey, we use the formula $V=\frac{d}{t}$. This formula also allows us to calculate the distance traveled if we know the duration and average speed of the journey (d = Vt) or the duration if we know the distance traveled and the average speed $\left(t=\frac{d}{V}\right)$.

When carrying out this sort of calculation, be careful to always use matching units!

A. Example 1

Problem: A boat makes a journey of 28 km in 2 hours 20 minutes. What is its average speed?

Solution: The journey time is given in hours and minutes.

We begin by converting this time into hours: 60 minutes is equal to one hour, so 20 minutes is equal to $\frac{1}{3}$ h, and 2 h 20 min is equal to $\frac{7}{3}$ h (because $2+\frac{1}{3}=\frac{7}{3}$ ).

Next we apply the formula $V=\frac{d}{t}$ (d is in km and t is in h, so V will be in km/h).

We replace d and t with their values, and get: $V=\frac{28}{{\displaystyle \frac{7}{3}}}=28\times \frac{3}{7}=\frac{28\times 3}{7}=\frac{4\times 7\times 3}{7}=12$.

The journey is made at an average speed of 12 km/h.

B. Example 2

Problem: While cycling in the mountains, Martin travels 8 km up a hill, at an average speed of 8 km/h. He descends along the same route at an average speed of 32 km/h.
So what is his average speed for the round trip?

Solution: From the problem we can quickly work out that the ascent took an hour  and the descent took a quarter of an hour $\left(\frac{8}{32}=\frac{8}{8\times 4}=\frac{1}{4}\right)$; so the round trip lasted an hour and a quarter, or 1.25 h (it is easier to use tenths or hundredths of an hour than minutes).

The length of the round trip is 16 km (2 × 8 = 16).
To calculate the average speed of the total journey, we use the formula $V=\frac{d}{t}$ (d is in km and t is in h, so V will be in km/h). We replace d and t with their values, and get: $V=\frac{16}{125}=128$.

The average speed of the round trip is 12.8 km/h.

Important Note: The average speed of the round trip is not the average of the two average speeds (that would be equal to $\frac{32+8}{2}$, or 20 km/h).

II. Calculating a distance

Problem: The speed of light is about 300,000 km/second. It takes about 8 minutes for light to get from the Sun to Earth. What is the distance of the planet Earth from the Sun?

Solution: We should convert the time of 8 minutes into seconds: 1 minute is equal to 60 seconds, so 8 minutes is equal to 480 seconds (because 8 min × 60 sec/min = 480 sec);
use the formula d = Vt (V is in km/s and t is in s, so d will be in km). We replace V and t with their values, and get: d = 300,000 km/sec × 480 sec = 144,000,000 km.
The distance of Earth from the Sun is about 144 million kilometers.

III. Calculating a time

Problem: Emily is going for a 15 km walk. She wants to get back at 4:30 pm and plans to walk at an average speed of 4 km/h. At what time should she set out?

Solution: First we can work out the journey time using the formula $\left(t=\frac{d}{V}\right)$ (d is in km and V is in km/h, so t will be in h); we replace V and d with their values, and get: $t=\frac{15}{4}=3+\frac{3}{4}$, so the walk will last three and three-quarter hours.
Now we can calculate Emily’s departure time. For ease of subtraction, we will convert the clock time 4:30 pm into the following notation: 16 hours 30 minutes.

Since $\frac{3}{4}$ h is equal to 45 min, we must carry out the following subtraction: 16 h 30 min – 3 h 45 min, which is the same as the 15 h 90 min – 3 h 45 min. We get the result 12 h 45 min.

Emily must leave at 12:45 pm (or a quarter to one) to get back at 4:30 pm.