An athlete, in a race, accelerates to a speed of 12.4 meters per second
in 3 seconds.
He runs at this speed for the next 5 seconds and slows down over the
last 2 seconds as shown in the speed-time graph above.
He crosses the finish line after 10 seconds.
The total distance covered is 100 m.
( b ) Calculate his speed when he crosses the finish line.
To calculate the athlete's speed when he crosses the finish line, we need to first calculate the total distance covered during each phase of his run.
1. Acceleration phase (first 3 seconds):
Initial speed, u = 0 m/s
Final speed, v = 12.4 m/s
Time, t = 3 seconds
Using the formula for acceleration: v = u + at
12.4 = 0 + a(3)
a = 12.4/3
a ≈ 4.13 m/s^2
Using the formula for distance covered during acceleration: s = ut + (1/2)at^2
s = 0(3) + (1/2)(4.13)(3)^2
s ≈ 18.675 m
2. Constant speed phase (next 5 seconds):
Speed = 12.4 m/s
Distance = speed * time = 12.4 * 5 = 62 m
3. Deceleration phase (last 2 seconds):
Initial speed = 12.4 m/s
Final speed = 0 m/s
Time = 2 seconds
Using the formula for deceleration: v = u + at
0 = 12.4 - a(2)
a = 12.4/2
a = 6.2 m/s^2
Using the formula for distance covered during deceleration: s = ut + (1/2)at^2
s = 12.4(2) + (1/2)(-6.2)(2)^2
s = 24.8 - 6.2(2)
s = 24.8 - 24.8
s = 0 m
Therefore, the total distance covered during the entire race is:
18.675 + 62 + 0 = 80.675 m
Since the total distance covered is 100 m, the remaining distance that the athlete covers after the deceleration phase is:
100 - 80.675 = 19.325 m
This remaining distance was covered at a constant speed, so we can calculate the time taken to cover this distance:
Distance = speed * time
19.325 = speed * time
19.325 = 12.4 * time
time = 19.325 / 12.4
time ≈ 1.56 seconds
Adding this time to the total time of 10 seconds, we find the speed when he crosses the finish line:
time = 10 + 1.56 = 11.56 seconds
Therefore, his speed when he crosses the finish line is 12.4 m/s.