# Integrate to get Position function

#### lax1113

1. The problem statement, all variables and given/known data
If a car is going 30m/s and see's an accident 60 meters in front of him/her, at what deceleration must the car apply brakes in order to stop in time?

2. Relevant equations
s(t) = $$\int(v)t$$
v(t) = $$\int(a)t$$

3. The attempt at a solution
So I know the physics equation that I could use for this very easily (vf^2 = vo^2+2a \Delta (x)
But, for my calculus class I have to do it obvously with only calculus. I don't know why i can't get this, but I have a feeling it will be on my exam tomorrow so I need to know how to show the work.

I am setting it up that V(t) = at + Vot, then integrating to find s(t), but after this point, I have
s(t) = 1/2 a t^2 + vot +s(0) s(0) is 0 for this case because it is from the point of applying brakes.... Anyway, where do I go from here? I am kinda lost and really don't know why.

Any help greatly appreciated, thanks!

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#### HallsofIvy

Okay, since you were initially going at 30 m/s, you have $v(t)= at+ 30$ and $s(t)= (1/2)at^2+ 30t$.

The point is that you want to have v(t)= 0 before s(t)= 60. Since the problem asks for a single answer, you want to stop just in time: v(t)= 0 and s(t)= 60. Solve those two equations for a.

#### lax1113

Halls,
Thank you very much for that. I don't know why I just couldn't see it right, I took physics more recently than calculus, so I couldn't remember exactly how to derive the equations.

By the way, a question very similar to this was on my final and i got it! thanks again!

Ben

"Integrate to get Position function"

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