# How to determine the total time of a runner if time is given for only a fraction of the entire race?

The runner's acceleration is constant. He covers the last 1/9 of the race in 6 seconds. How long does it take him to run the entire race?

I am stuck because there is such little information. I am not given the total distance, acceleration, or final velocity.

Using the position formula x=x(i)+v(i)t+1/2(a)(t^2) I find that t=sqrt(2x/a).

If I differentiate the function, then f'(t)=at, which gives the instantaneous velocity at t.

I'm not sure how to tackle this problem.

### 2 Answers

- AenimaLv 68 years agoFavorite Answer
You'll need to find equations that link the states between the last 1/9 distance of the race, the first 8/9 and the total length together.

We aren't given a distance, but call the total distance L or D.

For the first 8/9 ths you can use, vf^2 = v0^2 + 2a(x-x0)

Vf here would correspond to the initial speed between the last 1/9 of the length.

Vf = v0 + at

And you can use the first equation again but for the entire length:

Vff^2 = v0^2 + 2a(x-x0)

You can solve for acceleration with all these and then for total time using:

X = x0 + v0t + .5at^2 for the entire race.

I tried it myself and it was a bit challenging with all the algebra. But I got a final answer of 104.9 seconds. Is the answer given?

- fortadoLv 44 years ago
That is gonna be like spinning the bottle to peer who will get the crown, I say which ever one has the most fanatics from R/S. Every person there seems to have it out for the each of ya.~~Peace~~