Final answer:
The sprinter's time for the race, if they accelerate for 20 m and then maintain that velocity for the remaining 100 m, is approximately 9.78 seconds.
Step-by-step explanation:
To find the time for the race, we need to break it down into two parts: the time it takes to accelerate to the maximum speed and the time it takes to cover the remaining distance at that speed.
First, let's find the time it takes to accelerate to the maximum speed. We are given a rate of acceleration and a distance of 20 m. We can use the equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance. Plugging in the values, we get:
v^2 = (11.5 m/s)^2 + 2(0.500 m/s^2)(20 m)
v^2 = 132.25 m^2/s^2 + 20 m^2/s^2
v^2 = 152.25 m^2/s^2
v ≈ 12.34 m/s
Using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time, we can find the time it takes to accelerate:
12.34 m/s = 11.5 m/s + (0.500 m/s^2)t
0.840 m/s = (0.500 m/s^2)t
t ≈ 1.68 s
Now, let's find the time it takes to cover the remaining distance at the maximum speed. We know the distance is 100 m and the speed is 12.34 m/s. We can use the equation v = s/t, rearranged to solve for t:
t = s/v = 100 m / 12.34 m/s ≈ 8.10 s
Adding the time it takes to accelerate (1.68 s) to the time it takes to cover the remaining distance (8.10 s), we find the total time for the race:
Total time = 1.68 s + 8.10 s ≈ 9.78 s
Therefore, the sprinter's time for the race is approximately 9.78 seconds.