6.2 Quantity Represented by the Area under a Graph (Part 2)

Combination of Graphs

Example 2:

The diagram above shows the speed-time graph of a moving object for 15 seconds.

(a) State the length of time, in s, that the particle moves with constant speed.

(b) Calculate the rate of change of speed, in ms^-2, in the first 3 seconds.

(c) Calculate the average speed of the object in 15 seconds.

Solution:

(a)

Length of time that the particle moves with constant speed

= 9 – 3 = 6 s

(b)

Rate of change of speed in the first 3 seconds

= acceleration = gradient

\begin{array}{l} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \\ = \frac{6 - 3}{3 - 0} \\ = 1 m s^{- 2} \end{array}

(c)

Total distance travelled of the object in 15 seconds

= Area under the graph in the 15 seconds

= Area P + Area Q + Area R

\begin{array}{l} = [\frac{1}{2} (3 + 6) \times 3] + [(9 - 3) \times 6] + [\frac{1}{2} (15 - 9) \times 6] \\ = 13.5 + 36 + 18 \\ = 67.5 m \end{array}

Average speed of the object in 15 seconds

\begin{array}{l} = \frac{Total distance travelled}{Total time taken} \\ = \frac{67.5}{15} = 4.5 m s^{- 1} \end{array}