6. The speed of a car as a function of time is shown in the given figure. Find the distance travelled by the car in 8 seconds and its acceleration.
(The speed–time graph is a straight line starting from the origin and reaching a speed of \(20 \, \text{m/s}\) at \(8 \, \text{s}\).)
Solution
From the graph:
Initial speed, \(u = 0 \, \text{m/s}\)
Final speed, \(v = 20 \, \text{m/s}\)
Time taken, \(t = 8 \, \text{s}\)
Distance travelled in 8 seconds
Distance travelled is equal to the area under the speed–time graph.
The graph forms a triangle.
\[
\text{Distance} = \frac{1}{2} \times \text{base} \times \text{height}
\]
\[
= \frac{1}{2} \times 8 \times 20 = 80 \, \text{m}
\]
\text{Distance} = \frac{1}{2} \times \text{base} \times \text{height}
\]
\[
= \frac{1}{2} \times 8 \times 20 = 80 \, \text{m}
\]
Acceleration of the car
Acceleration is given by the slope of the speed–time graph:
\[
a = \frac{v – u}{t}
\]
\[
= \frac{20 – 0}{8} = 2.5 \, \text{m/s}^2
\]
\[
a = \frac{v – u}{t}
\]
\[
= \frac{20 – 0}{8} = 2.5 \, \text{m/s}^2
\]
Distance travelled in 8 s = \(80 \, \text{m}\)
Acceleration of the car = \(2.5 \, \text{m/s}^2\)
Acceleration of the car = \(2.5 \, \text{m/s}^2\)