Average Speeds

In the chapter on Travel Graphs, it was suggested that there was a relation between the slope of the graph at some point in the journey and the current speed of the traveller.

In this graph showing, we see that the traveller covered $40$40 km in the first $30$30 minutes. At this rate, the traveller would cover $80$80 km in a full hour. So, the speed in the first section was $80$80 km/h.

In the next $15$15 minutes, no distance was covered. So, the speed was zero. 

In the final section, the distance went from $40$40 km to $90$90 km and $30$30 minutes elapsed. The traveller covered $50$50 km in half an hour. This is equivalent to a speed of $100$100 km/h.

In reality, a journey does not usually consist of just a few sections each with a constant speed. We would expect a real journey to have fluctuating speeds so that the time-distance graph would look more like the following.

If we compare the two graphs above, we see that both start at the point $(0,0)$(0,0) - no distance covered at time zero. Then, both graphs pass through the points $(30,40)$(30,40), $(45,40)$(45,40) and $(75,90)$(75,90). 

Although in the second graph the small variations in the speed are shown by deviations from the straight line, we can think of the average speed between pairs of points in the journey, as though the graph between these points was indeed a straight line. We ignore the changes of speed along the way and consider only the overall elapsed time and the distance between points on the path.

Example

I left home at $8:45$8:45 a.m. and travelled the $12$12 kilometres to my destination, arriving at $9:05$9:05 a.m. What was my average speed?

The elapsed time for the journey was $20$20 minutes, which is $\frac{1}{3}$13​ of an hour. In that time I travelled $12$12 kilometres. It is clear that at this average rate I would have travelled $36$36 kilometres in a full hour. So, my average speed was $36$36 km/h. 

This calculation is equivalent to using the formula $\text{average speed}=\frac{\text{distance travelled}}{\text{elapsed time}}$average speed=distance travelledelapsed time​

Worked Examples

Question 1

Question 2

Question 3