An athlete, in a race, accelerates to a speed of 12.4 meters per second

Question

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.

Answer 1

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.