How Far Does a Sprinter Run During Acceleration in a 100m Dash?

[nathan scott]

Homework Statement

In the 100 m dash a sprinter accelerates from rest to a top speed with an acceleration whose magnitude is 2.68 m/s². After reaching top speed, he runs remainder of race with constant velocity. If total race is run in 12.0s, how far does he run during the acceleration phase?

Homework Equations

d₁= 1/2*2.68 m/s²*t²
d₂= vt = 2.68 m/s²*t*12.0s - t

The Attempt at a Solution

Not knowing if the above equations are correct I tried d₁+d₂=100m.
I multiplied everything, moved the 100m to the other side and tried the quadratic formula. I don't think I came out with the correct answer. I got approx. t= 3.5 or t= about 18. Any suggestions

 

[nathan scott]

Nevermind. I've finally figured out my errors.

 

[nlightner]

?

First, it is important to note that the equations provided are not entirely correct. The first equation, d1= 1/2*2.68 m/s2*t2, is missing a term for initial velocity and should be d1= 1/2*a*t^2. The second equation, d2= vt = 2.68 m/s2*t*12.0s - t, is incorrect and should be d2= v*t, where v is the constant velocity during the second phase of the race.

To solve this problem, we can use the equation d = v0*t + 1/2*a*t^2, where d is the distance, v0 is the initial velocity, t is the time, and a is the acceleration. In this case, we know that the initial velocity is 0 m/s and the acceleration is 2.68 m/s^2. We also know that the total time is 12.0 seconds and we want to find the distance during the acceleration phase, which we can call d1.

Plugging in these values, we get d1 = 0 + 1/2*2.68 m/s^2*(12.0 s)^2 = 161.28 m. This means that the sprinter runs 161.28 meters during the acceleration phase. To find the distance during the constant velocity phase, we can simply subtract this from the total distance, which is 100 m. Therefore, the distance during the constant velocity phase is 100 m - 161.28 m = 38.72 m.

In summary, the sprinter runs 161.28 meters during the acceleration phase and 38.72 meters during the constant velocity phase, for a total distance of 200 meters. It is always important to carefully check your equations and make sure they are correct before attempting to solve a problem.