# Set up Differential Equation for the following situation

1. Apr 16, 2012

### Gooolati

1. The problem statement, all variables and given/known data
Going at top speed, grand prix driver X leads archrival Y by
a steady 3 miles. Only 2 miles from the nish, X runs out of gas. Thereafter, X decelerates with
time at a rate proportional to the square of her (instantaneous) speed and in the next mile X's speed
exactly haves. Who wins? (Note : The course for this race is straight.)

2. Relevant equations
You need to set up a di erential equation based on what is described above.
There are some constants that you need to gure out. To see who wins, you need to check how long
does it take for each of the drivers to nish the race. Whoever needs less amount of time than the
other wins. Keep in mind that both drivers were going at top speed v0 and the velocity of the driver
X decreases as t increases, so we can use the same technique from the previous problem. Since
logarithm pops out, you need to use a calculator to solve the problem at the end. For instance, let
x(t) and y(t) be the positions of X and Y at time t, respectively. Set the nish to be the origin. If
they are approaching the origin from the left, then y(0) = 5, x(0) = 2. And it is obvious that
y(t) = 5 +v0t since v0 is the top speed. Note that x
0
is the velocity of X, and x
00
is the acceleration
or deceleration of X depending of its sign.

3. The attempt at a solution

I know that y(0)=-5 + v0(t) for all time t
At time t=0 x(0)=-2+v0(t) but after that time the car is decelerating at a rate of -v^2

So would x(t)=-2+v0(t)-(v^2)(t)?
I also know that when x is at -1 that it's speed at that point is 1/2 of its top speed, but I am confused about how that comes into play.
All help is appreciated. Thanks!