Kinematics Sprinter Question

[SuperCass]

Homework Statement

A world-class sprinter accelerates to his maximum speed in 3.8 s. He then maintains this speed for the remainder of a 100-m race, finishing with a total time of 10.4 s.

(a) What is the runner's average acceleration during the first 3.8 s?

(b) What is his average acceleration during the last 6.6 s?

(c) What is his average acceleration during the entire race?

(d) State why your answer to part (c) is not the average of the answers to parts (a) and (b).

Homework Equations

Kinematic Equations

The Attempt at a Solution

I know that part b is 0 because the runner maintains a constant velocity. But for parts a, c, and d, I'm not sure what to do. I know I have to find the distances he runs while his speed changes and when it is constant and then find the acceleration, but how do I do this?


Thank you so much for your help!

 

[rl.bhat]

Assume a as acceleration and v as the maximum velocity.
Using the kinematic equations
Find v(max) in terms of a and t1
Find x1 in terms of a and t1.
In the second part the velocity is constant.
So x2 = v(max)*t2.
Then put x1 + x2 = 100 m, and solve for a and v(max).

 

[tomcue]

To solve this problem, we can use the kinematic equations to calculate the runner's acceleration. Let's start with part (a), where we need to find the average acceleration during the first 3.8 seconds of the race. We can use the following kinematic equation:

v = u + at

where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. We know that the final velocity is the maximum speed the runner reaches, which we can calculate by dividing the distance (100 m) by the total time (10.4 s):

v = 100 m / 10.4 s = 9.62 m/s

Since the runner starts from rest, the initial velocity (u) is 0 m/s. We also know that the time (t) is 3.8 s. Plugging these values into the equation, we get:

9.62 m/s = 0 m/s + a(3.8 s)

Solving for a, we get a = 2.53 m/s^2. This is the average acceleration during the first 3.8 seconds of the race.

For part (b), we know that the average acceleration during the last 6.6 seconds is 0 m/s^2, as the runner maintains a constant velocity. This means that there is no change in his speed during this time.

Now, to find the average acceleration during the entire race (part c), we can use the average acceleration formula:

a = (vf - vi) / t

where vf is the final velocity, vi is the initial velocity, and t is the total time. We already know that vf = 9.62 m/s and vi = 0 m/s. Plugging these values into the equation, we get:

a = (9.62 m/s - 0 m/s) / 10.4 s = 0.92 m/s^2

Therefore, the average acceleration during the entire race is 0.92 m/s^2.

Now, you may notice that this answer is not the average of the answers for parts (a) and (b). This is because the runner's acceleration is not constant throughout the race. In part (a), the runner is accelerating to reach his maximum speed, while in part (b), he is maintaining a constant speed. Averaging these two values would not give