Answer:
a) To find the runner's acceleration as they speed up from rest, we can use the equation:
v^2 = u^2 + 2as
Where:
v is the final velocity (9.0 m/s),
u is the initial velocity (0 m/s),
a is the acceleration (unknown),
and s is the displacement (25 m).
Plugging in the values, we have:
(9.0 m/s)^2 = (0 m/s)^2 + 2a(25 m)
81 m^2/s^2 = 0 + 50a
81 m^2/s^2 = 50a
To solve for a, we divide both sides by 50:
a = 81 m^2/s^2 / 50
a ≈ 1.62 m/s^2
Therefore, the runner's acceleration as they speed up from rest is approximately 1.62 m/s^2.
b) To find the time it took for the runner to reach their maximum speed, we can use the equation:
v = u + at
Where:
v is the final velocity (9.0 m/s),
u is the initial velocity (0 m/s),
a is the acceleration (1.62 m/s^2, as calculated in part a),
and t is the time (unknown).
Plugging in the values, we have:
9.0 m/s = 0 + (1.62 m/s^2)t
Simplifying the equation:
9.0 m/s = 1.62 m/s^2 * t
To solve for t, we divide both sides by 1.62:
t = 9.0 m/s / 1.62 m/s^2
t ≈ 5.56 seconds
Therefore, it took approximately 5.56 seconds for the runner to reach their maximum speed.
c) To find the time it took for the runner to run the entire race, we need to consider the time it took for the acceleration phase (part b) and the time it took to cover the remaining distance at the constant velocity.
The time for the acceleration phase was found to be approximately 5.56 seconds.
To find the time for the remaining distance, we can use the equation:
s = vt
Where:
s is the remaining distance (75 meters, calculated as 100 meters - 25 meters),
v is the constant velocity (9.0 m/s),
and t is the time for the remaining distance (unknown).
Plugging in the values, we have:
75 m = 9.0 m/s * t
To solve for t, we divide both sides by 9.0:
t = 75 m / 9.0 m/s
t ≈ 8.33 seconds
Therefore, it took approximately 8.33 seconds for the runner to cover the remaining distance at their constant velocity.
To find the total time for the entire race, we add the time for the acceleration phase and the time for the remaining distance:
Total time = Time for acceleration + Time for remaining distance
Total time = 5.56 seconds + 8.33 seconds
Total time ≈ 13.89 seconds
Therefore, it took approximately 13.89 seconds for the runner to complete the entire race.
