A runner hopes to complete a 10,000 m race in a time of 30
minutes. After running at a constant speed for exactly 27 minutes,
there are still 1100 m to go. The runner then accelerates at
0.2 m/s2 for how many seconds in order to achieve the desired
time?
The runner wants to go the remaining 1100 m in 3 minutes. That will require an average speed of 6.11 m/s. His speed for the first 2700 minutes was
8900m/(27*60 s) = 5.49 m/s.
If he accelerates for time T (seconds) and then maintains constant speed for 180-T seconds,
Final speed Vf = 5.49 + 0.2 T
Distance travelled in final 3 minutes
1100 = Vf*(180-T) + (Vf + 5.49)T/2
Substitute 5.49 +0.2T for Vf in the final equation and solve for T.
12.4
To find out how many seconds the runner needs to accelerate for in order to achieve the desired time, we need to consider the runner's initial speed and the distance they still have to cover.
We know that after running for 27 minutes (or 1620 seconds), the runner still has 1100 m to go. This means that they have covered a distance of 10,000 m - 1100 m = 8900 m in 27 minutes.
To find the initial speed of the runner, we can calculate the average speed over the first 27 minutes. Average speed is distance divided by time, so:
Average speed = 8900 m / 1620 s
The runner's initial speed is the average speed over the first 27 minutes.
Now, let's determine the initial speed:
Initial speed = 8900 m / 1620 s
To find the remaining time needed to complete the race, we subtract the time already elapsed (27 minutes or 1620 seconds) from the desired race time (30 minutes or 1800 seconds):
Remaining time = 1800 s - 1620 s
Next, we need to calculate the distance covered during the acceleration period. To do this, we use the formula:
Distance = Initial speed * Time + (1/2) * Acceleration * Time^2
Since we know the initial speed, acceleration, and remaining time, we can solve for the distance covered during acceleration.
1100 m = Initial speed * Time + (1/2) * 0.2 m/s^2 * Time^2
Simplifying the equation:
1100 m = (Initial speed + 0.1 m/s^2 * Time) * Time
This is a quadratic equation, and we can solve it using the quadratic formula:
Time = (-b + sqrt(b^2 - 4ac)) / 2a
Where a = 0.1, b = Initial speed, and c = -1100.
Plug in the values and calculate the time.