Answer:
a) The acceleration of Usain Bolt was 2.98 m/s².
His maximum speed was (v = 2.98 m/s² · 3.00 s) 8.94 m/s.
b) The acceleration was 3.21 m/s² and the maximum speed was 9.63 m/s
Explanation:
The equation for the position of Bolt during the first 3 s is as follows:
x = x0 + 1/2 · a · t²
Where:
x = position at time t
a = acceleration
t = time
x0 = initial position
After that time, the position is:
x = x0 + v · t
Where "v" is the velocity
During the acceleration, the velocity is:
v = a · t
The maximum speed reached by Bolt is then:
v = a · 3.00 s
The distance traveled during the first 3 seconds is:
x = x0 + 1/2 · a · t² (x0 = 0)
x = 1/2 · a · (3.00 s)²
x = 4.50 s² · a
Then, in the equation of the position after the first 3 s, we can replace the initial position for the position traveled during the first 3 s and the velocity for the velocity reached in those 3 s.
x = x0 + v · t
x0 = 4.50 s² · a
v = a · 3.00 s
Then:
x = 4.50 s² · a + 3.00 s · a ·t
At t = 9.69 s, x = 100 m. Then:
100 m = 4.50 s² · a + 3.00 s · a · 9.69 s
100 m = 33.6 s² · a
100 m / 33.6 s² = a
a = 2.98 m/s²
The acceleration of Usain Bolt was 2.98 m/s².
His maximum speed was (v = 2.98 m/s² · 3.00 s) 8.94 m/s.
b) The calculations are the same only that in the final step we have to use x = 200 m instead of 100 m and t = 19.30 s instead of 9.69 s:
200 m = 4.50 s² · a + 3.00 s · a · 19.30 s
200 m = 62.4 s² · a
200 m / 62.4 s² = a
a = 3.21 m/s²
v = 3.21 m/s² · 3.00 s = 9.63 m/s
The acceleration was 3.21 m/s² and the maximum speed was 9.63 m/s
