Average Speeds

Lesson

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

Maria travels by car for $420$420 km. The trip takes $10$10 hrs. What is the average speed of the trip?

Question 2

With respect to the following graph, what was the:

1. Total time taken for the journey?

2. Total distance covered in the journey?

3. Average speed during the journey? Answer to two decimal places.

Question 3

Dave states that his trip between Amsterdam and Moscow had an average speed of $80$80 km/h. Use the table, which shows the distances between cities in kilometres, to find how long the trip from Amsterdam to Moscow took Dave.

 Berlin London Copenhagen Amsterdam Moscow Rome Warsaw Berlin 587 613 354 2133 1301 1086 London 587 362 925 1601 1175 515 Copenhagen 613 362 950 1562 1526 668 Amsterdam 354 925 950 2479 1433 1432 Moscow 2133 1601 1562 2479 2354 1143 Rome 1301 1175 1526 1433 2354 1304 Warsaw 1086 515 668 1432 1143 1304

Outcomes

MS1-12-3

interprets the results of measurements and calculations and makes judgements about their reasonableness