Three runners run a 10 metre race. Each runner runs according to the following rules:
Runner 1: s = 2.4 + 0.75t
Runner 2: s = 0.2t(t - 5.1)
Runner 3: s = 0.2t(t - 5.1)(t - 9.1)
Where displacement (s) is in metres and time (t) is in minutes.
Calculate for each runner: their starting position, any changes in direction over the race (when and where it happened) and their finishing time for the race
To calculate the starting position, changes in direction over the race, and finishing time for each runner, we need to analyze the given equations for their displacement with respect to time.
1. Runner 1: s = 2.4 + 0.75t
Starting Position: The starting position is given by the equation when t = 0.
s = 2.4 + 0.75(0) = 2.4 meters
There are no changes in direction for Runner 1. Since the equation only contains a linear term (0.75t), there are no turning points.
Finishing Time: To find the finishing time, we need to determine when s = 10 meters.
10 = 2.4 + 0.75t
7.6 = 0.75t
t ≈ 10.1 minutes (rounded to one decimal place)
2. Runner 2: s = 0.2t(t - 5.1)
Starting Position: The starting position is given by the equation when t = 0.
s = 0.2(0)(0 - 5.1) = 0 meters
Changes in Direction: To find changes in direction, we need to find when the velocity (the derivative of displacement with respect to time) equals zero.
First derivative: v = ds/dt = 0.2(2t - 5.1)
0 = 0.2(2t - 5.1)
0 = 2t - 5.1
2t = 5.1
t ≈ 2.6 minutes (rounded to one decimal place)
There is a change in direction around 2.6 minutes, where the runner starts changing direction from positive to negative displacement.
Finishing Time: To find the finishing time, we need to determine when s = 10 meters.
10 = 0.2t(t - 5.1)
0 = 0.2t^2 - 1.02t + 10
Using the quadratic formula: t = [-b ± sqrt(b^2 - 4ac)] / (2a)
t ≈ 10.9 minutes (rounded to one decimal place)
3. Runner 3: s = 0.2t(t - 5.1)(t - 9.1)
Starting Position: The starting position is given by the equation when t = 0.
s = 0.2(0)(0 - 5.1)(0 - 9.1) = 0 meters
Changes in Direction: To find changes in direction, we need to find when the acceleration (the second derivative of displacement with respect to time) equals zero.
First derivative: v = ds/dt = 0.2(3t^2 - 28.8t + 46.41)
Second derivative: a = dv/dt = 0.2(6t - 28.8)
0 = 0.2(6t - 28.8)
0 = 6t - 28.8
6t = 28.8
t = 4.8 minutes
There is a change in direction around 4.8 minutes, where the runner starts changing direction from negative to positive displacement.
Finishing Time: To find the finishing time, we need to determine when s = 10 meters.
10 = 0.2t(t - 5.1)(t - 9.1)
0 = 0.2t^3 - 4.4t^2 + 23.0t - 46.41
Using numerical methods or a graphing calculator, the solution is t ≈ 11.9 minutes.
To recap:
Runner 1: Starting Position = 2.4 meters, No changes in direction, Finishing Time ≈ 10.1 minutes
Runner 2: Starting Position = 0 meters, Change in direction at 2.6 minutes, Finishing Time ≈ 10.9 minutes
Runner 3: Starting Position = 0 meters, Change in direction at 4.8 minutes, Finishing Time ≈ 11.9 minutes