A runner hopes to complete the 10,000m run in less than 30 minutes. After exactly 27 minutes, there are still 1100m to go. The runner must then accelerate at 0.20 m/s^2 for how many seconds in order to achieve the desired time?
To find the time required to accelerate and complete the remaining distance, we can use the formula:
s = ut + (1/2)at^2,
where s is the distance, u is the initial velocity, t is the time, and a is the acceleration.
Given that the remaining distance is 1100m and the initial velocity is unknown, we need to determine the initial velocity (u) using the information given.
We know that the runner has completed 9700m (10,000m - 1100m) in 27 minutes, which is 27 * 60 = 1620 seconds.
So, the initial velocity (u) can be found using the formula:
u = s / t = 9700m / 1620s ≈ 5.99 m/s.
Now, we can find the time (t) required to accelerate and cover the remaining distance.
Using the formula:
s = ut + (1/2)at^2,
where s = 1100m, u = 5.99 m/s, and a = 0.20 m/s^2,
we rearrange the formula to solve for t:
1100m = 5.99 m/s * t + (0.5) * 0.20 m/s^2 * t^2.
Rearranging and simplifying the equation:
0.1t^2 + 5.99t - 1100 = 0.
Now, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a.
Solving for t:
t = (-5.99 ± √(5.99^2 - 4 * 0.1 * -1100)) / 2 * 0.1.
Using the positive root of the equation:
t = (-5.99 + √(5.99^2 - 4 * 0.1 * -1100)) / 2 * 0.1 ≈ 5.08 seconds.
Therefore, the runner must accelerate for approximately 5.08 seconds in order to achieve the desired time of completing the 10,000m run in less than 30 minutes.