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The distance-time graph shows the distance travelled by a car from 1400 to 1700 hours. (a) Calculate the speed at 1630 hours. (b) Represent the distance-time graph above on a speed-time graph.

Mathematics

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Answer

a

15 km/h15 \text{ km/h}

Step-by-step solution

5 steps

Part a· 3 steps

Calculate the speed at 16:30 hours.

  1. Understand that speed on a distance-time graph is represented by the gradient of the line.

    The time 16:30 hours falls within the segment of the graph from 15:00 to 17:00.

  2. Identify the coordinates for this segment.

    At t1=15:00t_1 = 15:00 hours, the distance d1=60 kmd_1 = 60 \text{ km}.
    At t2=17:00t_2 = 17:00 hours, the distance d2=90 kmd_2 = 90 \text{ km}.

  3. Calculate the speed (gradient) for this segment.

    Speed=Change in distanceChange in time=d2−d1t2−t1\text{Speed} = \frac{\text{Change in distance}}{\text{Change in time}} = \frac{d_2 - d_1}{t_2 - t_1} Speed=90 km−60 km17:00−15:00\text{Speed} = \frac{90 \text{ km} - 60 \text{ km}}{17:00 - 15:00} Speed=30 km2 hours\text{Speed} = \frac{30 \text{ km}}{2 \text{ hours}} Speed=15 km/h\text{Speed} = 15 \text{ km/h}

    Since this segment is a straight line, the speed is constant throughout this interval.

    The speed at 16:30 hours is 15 km/h\boxed{15 \text{ km/h}}.

Part b· 2 steps

Represent the distance-time graph above on a speed-time graph.

  1. Analyze the speed for each segment of the distance-time graph.

    Segment 1 (from 14:00 to 15:00):
    Time interval = 15:00−14:00=1 hour15:00 - 14:00 = 1 \text{ hour}.
    Distance covered = 60 km−0 km=60 km60 \text{ km} - 0 \text{ km} = 60 \text{ km}.
    Speed = 60 km1 hour=60 km/h\frac{60 \text{ km}}{1 \text{ hour}} = 60 \text{ km/h}.

    Segment 2 (from 15:00 to 17:00):
    Time interval = 17:00−15:00=2 hours17:00 - 15:00 = 2 \text{ hours}.
    Distance covered = 90 km−60 km=30 km90 \text{ km} - 60 \text{ km} = 30 \text{ km}.
    Speed = 30 km2 hours=15 km/h\frac{30 \text{ km}}{2 \text{ hours}} = 15 \text{ km/h}.

  2. Plot these constant speeds on the speed-time graph.

    The x-axis of the speed-time graph represents time in hours, starting from 0 (which corresponds to 14:00).

    • From t=0t=0 hours (14:00) to t=1t=1 hour (15:00), the speed is 60 km/h60 \text{ km/h}. Draw a horizontal line at 60 km/h60 \text{ km/h} from t=0t=0 to t=1t=1.
    • From t=1t=1 hour (15:00) to t=3t=3 hours (17:00), the speed is 15 km/h15 \text{ km/h}. Draw a horizontal line at 15 km/h15 \text{ km/h} from t=1t=1 to t=3t=3.

    The speed-time graph will look like this:

    \begin{tikzpicture}[scale=0.8] \draw[->] (0,0) -- (4.5,0) node[below] {time (hours)}; \draw[->] (0,0) -- (0,4.5) node[left] {speed (km/h)}; \foreach \x in {0,1,2,3,4} \draw (\x,0) node[below] {\x}; \foreach \y in {0,20,40,60,80} \draw (0,\y/20*4) node[left] {\y}; \draw[dashed, gray] (0,1.2) -- (4,1.2); % 15 km/h line \draw[dashed, gray] (0,2.4) -- (4,2.4); % 30 km/h line \draw[dashed, gray] (0,3.6) -- (4,3.6); % 60 km/h line \draw[thick, blue] (0,3.6) -- (1,3.6); % First segment: 60 km/h for 1 hour \draw[thick, blue] (1,0.9) -- (3,0.9); % Second segment: 15 km/h for 2 hours \draw[fill=black] (0,3.6) circle (1.5pt); \draw[fill=black] (1,3.6) circle (1.5pt); \draw[fill=black] (1,0.9) circle (1.5pt); \draw[fill=black] (3,0.9) circle (1.5pt); \end{tikzpicture}

    The graph should show a horizontal line at 60 km/h60 \text{ km/h} from t=0t=0 to t=1t=1 hour, and another horizontal line at 15 km/h15 \text{ km/h} from t=1t=1 to t=3t=3 hours.

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