I made this problem up as I was driving to school. Suppose you press the gas pedal on your car and your rpm's rev from 800 to 6200. You also know that your car's max horsepower is 150 (at 6200rpm). Find avg acceleration and acceleration function of the car as a function of RPMs. Time is defined in seconds as the time it takes to get from 800 to 6200rpms. For simplicity we will assume a gear ratio of 1, and of course neglect friction. I know this can be done using kinematic equations and a few functions but I wanted to do it this way and this is what I got: P = W/t For every infinitesimally small increment of RPM the engine produces a power P in an infinitesimally small time dt or, dW = P(t) dt Assuming a quadratic relationship between power and time, we'll say P = c*t^2 where c is some proportionality constant. dW = c*t^2 dt and W = c/3*t^3 Work is F*d and F = m*a and the equation becomes 3mad = ct^3 and solved for a is: a(t) = (ct^3)/(3md) I dont have maple to graph the function to see if it makes sense, but a cubic function isnt really what I was expecting. So with a linear relationship between power and time we get: a(t) = ct^2/(2md) which still doesnt sound right. Something tells me I should be integrating with respect to Power, im not sure though. All I can get for certain in my head is that at a certain RPM constant power is produced. Constant power means constant jerk (skipping some kinematics here). So variable power should mean constant jounce (whatever the 4th derivative of position is) and quadratic acceleration.