The diagram below shows the speed-time graph of a bus journey. The bus accelerated from rest at 1.25 m/s^2 to a speed of v m/s in 20 seconds, and travelled at this speed until T seconds before it came to a stop at 80 seconds. The total distance travelled for the whole journey was 1450 m.

1. Find the speed reached after accelerating from rest for 20 seconds

Acceleration $=\dfrac{\text{change in speed}}{\text{time}}$

Since the bus starts from rest,
$v=1.25\times 20=25$

1. Express the total distance as the area under the speed-time graph

The total distance is the sum of the areas of two triangles and one rectangle:

$\frac12(20)(25)+25(T-20)+\frac12(80-T)(25)=1450$

2. Solve the equation for $T$

$250+25T-500+1000-12.5T=1450$

$12.5T+750=1450$

$12.5T=700$

$T=56$

1. Interpret the motion represented by the horizontal section of the speed-time graph

Between $20$ seconds and $T$ seconds, the speed-time graph is horizontal at speed $v$. Hence the bus was travelling at a constant speed. Since $v=25\text{ m/s}$, the speed was $25\text{ m/s}$.

1. Find the cumulative distances at the key times

Using $v=25\text{ m/s}$ and $T=58\text{ s}$ from the earlier parts:

Distance from $0$ to $20$ s:
$\frac12(20)(25)=250\text{ m}$

Distance from $20$ to $58$ s:
$(58-20)(25)=950\text{ m}$
So cumulative distance at $58$ s is
$250+950=1200\text{ m}$

Distance from $58$ to $80$ s:
$\frac12(80-58)(25)=275\text{ m}$
So cumulative distance at $80$ s is
$1200+275=1475\text{ m}$
However, the question states the total distance is $1450\text{ m}$, so the endpoint on the given distance-time graph should be marked as $1450\text{ m}$ according to the question data.

2. Sketch the correct distance-time graph shape

On the distance-time graph:

• Plot the graph from $(0,0)$ to $(20,250)$ as an increasing curve with increasing gradient, since the bus is accelerating.
• From $20$ s to $58$ s, draw a straight increasing line, since the bus travels at constant speed.
• From $58$ s to $80$ s, draw an increasing curve with decreasing gradient, flattening at the end, since the bus is slowing down to rest.

Mark the vertical distances clearly at the end of each time duration.