i need help in calculus,?

Question

Two drivers, Alison and Kevin, are participating in a drag race. Beginning from a standing start, they each proceed with a constant acceleration. Kevin covers the last 1/4 of the distance in 3 seconds, whereas Alison covers the last 1/3 of the distance in 4 seconds. Who wins and by how much time?

the answer to the question is; Alison wings by 6root(3) - 4 root(6) = 0.594 sec. but i dont know how they get the answer. i know the math, but i dont know how to set up the equation. plz help!

Answer 1

Ok. So let's set this one up.

1)

They both move in a constant acceleration.

If I say D(t) is a distant they move in time(t), then D''(t) = k. Second derivative of D(t) is acceleration, and it's constant, so k is an unknown constant which is an acceleration. Now Integrate this in order to get the speed function.

⌠ k dt = kt + C
⌡

So....D'(t) = kt + c. It's the speed function. However, c is an initial speed, which is zero. So, D'(t) = kt. Do you get this? They're not starting their race with initial speed. They're all stopped in the beginning, so the initial speed is 0.

Now. Let's get D(t), which is a distance function.


⌠ kt dt = 1/2k * t^2 + C
⌡

So D(t) = 1/2 kt^2 + c. However, the initial distance is 0. They don't start their race in the middle of the track. The start position is 0. so C is also 0 here. Thus...

D(t) = 1/2k * t^2


2)

This part is really disturbing. Ok, now. Let's talk about how Kevin did. I'll say "T" is a time took for Kevin to finish his race in seconds. Then we can make another equation for Integration.


⌠ T 　　 kt dt = 1/4 * ⌠ T kt dt
⌡ T - 3　　　　　　　　　⌡ 0


Do you get this? Integration of speed function, which is kt, from T to T-3 is the distance he drove in the last 3 seconds of his race. Integration from T to 0 is the distance he drove in his entire race. So the distance he drove in the last 3 seconds are 1/4th of the entire distance. You integrate that, and you get the following equation.

1/2k * T^2 - 1/2k * (T - 3)^2 = 1/4 * 1/2k * T^2

So.. Solve this in terms of Time. If you finished your Algebra class in high school, you'll have no prob solving that equation. I got 12+- 6root(3). There are two answers for Kevin, 12+6root(3) and 12-6root(3). However, 12-6root(3) does not qualify, because the time it took for him should be greater than 3 seconds. So, the time for Kevin is 12 + 6root(3).

3)

now, do the same thing with Alison.


⌠ T 　　 kt dt = 1/3 * ⌠ T kt dt
⌡ T - 4　　　　　　　　　⌡ 0


is equal to...

1/2k * T^2 - 1/2k * (T - 4)^2 = 1/6 * 1/2k * T^2



Solve this for T. and you get 12+4root(6). For the same reason, 12-4root(6) does not qualify, because it's less than 4 seconds. Ok. Now let's compare the result.

4)

Kevin : 12 + 6root(3) sec
Alison : 12 + 4root(6) sec

6root(3) is root(108)
4root(6) is root(48)

Thus, Alison took less time to finish the race. Alison won. By how much? Subtract the Kevin's time by Alison's time.

12 + 6root(3) - (12 + 4root(6)) = 6root(3) - 4root(6) = 0.594 sec.


Problem solved. I hope you understand this.

Answer 2

x=distance

let us compare their speed
v1=x/4*3
v2=x/3*4

compare v1 and v2
v2>v1
Alison wins