Final answer:
Usain Bolt takes approximately 3.43 seconds to reach his top speed of 12.7 m/s with an acceleration of 3.7 m/s². During this acceleration, he covers about 21.9 meters. To complete the 100m dash, it takes him an additional 6.15 seconds at top speed, totaling approximately 9.58 seconds for the race.
Step-by-step explanation:
To calculate the time it takes for Usain Bolt to run the 100m dash based on the given acceleration and top speed, we first need to determine the time it takes to reach that top speed. Using the formula for acceleration (a = Δv / Δt) and rearranging it to solve for time (Δt = Δv / a), with Δv being the change in velocity (top speed - initial speed, assuming the initial speed is 0), we find the time to accelerate to top speed to be Δt = 12.7 m/s / 3.7 m/s² = 3.43 s.
Next, we'll calculate the distance covered during this acceleration period using the formula s = ut + 0.5at², where u is the initial velocity (0 m/s), a is the acceleration (3.7 m/s²), and t is the time calculated earlier (3.43 s). This yields a total distance covered during acceleration of s=0.5×3.7 m/s²×3.43 s² ≈ 21.9 m. Since the distance covered during acceleration is 21.9 m, the remaining distance at his top speed is 100 m - 21.9 m = 78.1 m.
At his top speed of 12.7 m/s, time to cover the remaining distance is Δt = distance/speed = 78.1 m / 12.7 m/s ≈ 6.15 s. Therefore, the total time to complete the 100m dash is the time taken to reach top speed plus the time to cover the remaining distance, which is 3.43 s + 6.15 s ≈ 9.58 s.