1. The problem statement, all variables and given/known data 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. 2. Relevant equations Ke = (1/2)MV^2 d(Ke)dt = -m*g*sin(slope)v 3. 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: Keo = (.5)*mvo2. dKe/dt = -mgsin(∅)vo how can I relate these to t, take a derivative of the first equation and set it equal to the 2nd?