Car Coasting Up a Hill: Equations for Velocity and Height

[terwilld]

Homework Statement

This is not a homework question per se but rather something I was thinking about while driving the other day.

I am in a car on a hill with constant slope traveling at an initial speed. How can I come up with a set of equations describing both velocity as a function of time and and height as a function of time.

Homework Equations

Ke = (1/2)MV^2
d(Ke)dt = -m*g*sin(slope)v

The Attempt at a Solution

All attempts to find solutions to this question have been fruitless. Most physics / calc books seem to offer this question in a simplified version: how far up the hill does the car make it. I am far more interested in plotting the curve of speed as a function of time.

Based on my understanding:

As the car travels up the hill, kinetic energy (Ke) is converted to potential energy: Ke = Pe. Potential energy = mgh. The kink seems to be that the change in potential energy is not the same. At the start, speed is the greatest, thus the change in potential energy is greatest ( = mg(sin(slope)*v). However as the car climbs the hill, its speed changes, thus the rate at which its speed changes, changes.

so far:
Initial conditions:
Ke_o = (.5)*mv_o².
dKe/dt = -mgsin(∅)v_o

how can I relate these to t, take a derivative of the first equation and set it equal to the 2nd?

 

[Dick]

Your dKe/dt notion is not needed. -mgsin(∅)=F is, in fact, a force. It's the force slowing your car down. You can convert that into the rate at which you car is slowing down by using Newton's law, F=ma. So a=(-gsin(∅)) is the deceleration rate of your car. And it's a constant since the slope is constant. You really don't need energy at all to solve the problem you've got in mind.