The average speed from E to F = 300 − 200 _________
5 − 2
The average speed from F to G = 550 − 300 _________
7 − 5
= 100 ____
3 = 33 1 _
= 250 ____
3 km/h = slope of EF. 2 = 125 km/h = slope of FG.
The average speed between two points on the distance-time graph shown is the slope of the line joining the two points.
But what if the graph of distance against time is a curve rather than a series of straight lines?
WORKED EXAMPLE Dropping a stone
A stone drops from the top of a building at time t = 0. Its distance s in metres, t seconds after it drops, is given by s = 4·9 t 2 . Find its average speed between t = 2 and t = 5.
t (s) 012345
s (m) 04·919·644·178·4 122·5 s
100 120 140 Q(t2, s2) = (5, 122·5)
80
40 60
20 1
Average speed = Distance ________
Time The average speed = =
______ t2 − t1
s2 − s1
______ t2 − t1
s2 − s1
2 34 5 Time (s)
= 122·5 − 19·6 ___________
5 − 2 = 34·3 m/s = the slope of the line PQ.
In the worked example above, as in most realistic situations, the stone’s speed is changing continuously as it moves from point to point. The question arises as to what is the exact speed at any point. In other words, what is the instantaneous speed?
Instantaneous rate of change s (m)
The instantaneous rate of change of distance with respect to time is called instantaneous speed. The instantaneous speed of a body is its speed v at any instant of time t.
Instantaneous rate of change is usually just called rate of change. It is the slope m of the tangent k to the curve at any point on the curve
