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The diagram below is a speed-time graph for an athlete who took 100 seconds to complete a race. The athlete ran 1,75 km in 100 seconds. Calculate the value of V. and acceleration of the athlete from time 60 seconds to time 100 seconds.

Mathematics

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2 parts
a

25 m/s25 \text{ m/s}

b

−0.625 m/s2-0.625 \text{ m/s}^2

Step-by-step solution

5 steps

Here's the solution to question 29:

Part a· 3 steps

Calculate the value of VV.

  1. Convert the total distance from kilometers to meters.

    The athlete ran 1.75 km1.75 \text{ km} in 100 seconds100 \text{ seconds}.

    1.75 km=1.75×1000 m=1750 m1.75 \text{ km} = 1.75 \times 1000 \text{ m} = 1750 \text{ m}
  2. Calculate the total distance covered as the area under the speed-time graph.

    The graph is a trapezium. The area of a trapezium is given by 12×(sum of parallel sides)×height\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}.
    In this speed-time graph, the "height" of the trapezium is the maximum speed VV. The parallel sides are the time intervals on the x-axis.
    The top parallel side (where speed is constant) is from t=20 st=20 \text{ s} to t=60 st=60 \text{ s}, so its length is 60−20=40 s60 - 20 = 40 \text{ s}.
    The bottom parallel side (total time) is from t=0 st=0 \text{ s} to t=100 st=100 \text{ s}, so its length is 100−0=100 s100 - 0 = 100 \text{ s}.

    Total distance=12×(40 s+100 s)×V m/s\text{Total distance} = \frac{1}{2} \times (40 \text{ s} + 100 \text{ s}) \times V \text{ m/s} Total distance=12×(140 s)×V m/s\text{Total distance} = \frac{1}{2} \times (140 \text{ s}) \times V \text{ m/s} Total distance=70V m\text{Total distance} = 70V \text{ m}
  3. Equate the calculated total distance to the given total distance and solve for VV.

    70V=175070V = 1750 V=175070V = \frac{1750}{70} V=1757V = \frac{175}{7} V=25 m/sV = 25 \text{ m/s}

    The value of VV is 25 m/s\boxed{25 \text{ m/s}}.

Part b

−0.625 m/s2-0.625 \text{ m/s}^2

Calculate the acceleration of the athlete from time 60 seconds60 \text{ seconds} to time 100 seconds100 \text{ seconds}.

  1. Identify the coordinates for the given time interval.

    Acceleration is the gradient of the speed-time graph.
    At t1=60 st_1 = 60 \text{ s}, the speed is V1=V=25 m/sV_1 = V = 25 \text{ m/s}.
    At t2=100 st_2 = 100 \text{ s}, the speed is V2=0 m/sV_2 = 0 \text{ m/s}.

  2. Calculate the acceleration using the formula for gradient.

    Acceleration=change in speedchange in time=V2−V1t2−t1\text{Acceleration} = \frac{\text{change in speed}}{\text{change in time}} = \frac{V_2 - V_1}{t_2 - t_1} Acceleration=0 m/s−25 m/s100 s−60 s\text{Acceleration} = \frac{0 \text{ m/s} - 25 \text{ m/s}}{100 \text{ s} - 60 \text{ s}} Acceleration=−25 m/s40 s\text{Acceleration} = \frac{-25 \text{ m/s}}{40 \text{ s}} Acceleration=−58 m/s2\text{Acceleration} = -\frac{5}{8} \text{ m/s}^2 Acceleration=−0.625 m/s2\text{Acceleration} = -0.625 \text{ m/s}^2

    The acceleration is −0.625 m/s2\boxed{-0.625 \text{ m/s}^2}.

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