A car on a hill of constant slope accelerates from rest under gravity to a point on the hill. If you want to maximize the velocity of the car at this point, where along the length of the car (e.g. downhill or uphill) would you fix an additional mass?

Hint: What fundamental dynamics equation governs this situation?

(a) the finishing point is on the slope of the hill
(b) the finishing point is on a flat below the slope of the hill.

When looking at these type of problems, it’s important to start with our governing equation:

mgh = 1/2mv^2 + 1/2Iw^2

where:
Potential Energy = mgh
Translational Kinetic Energy = 1/2mv^2
Rotational Kinetic Energy =1/2Iw^2

As you travel down the slope, the potential energy term decrease as your translational kinetic energy and rotational kinetic energy increase. In the application of a car, a car has an abundance of mass that doesn’t rotate and only converts into translation kinetic energy (don’t get this confused in the case of a hollow or solid cylinder!). As such , rotational kinetic energy does not have a dominating effect on energy transferred from potential energy.

Since both cars weight same regardless of where the extra weight is attached, the cars reach the bottom of the hill at the same time. However, by fixing the mass to the back of the car (uphill), the center of the mass is higher up than if the weight were at the front of the car. By doing this, the car has more potential energy which translates into more kinetic energy (energy is conserved).

The car with the weight fixed on the back of the car will have more maximized velocity when reaching a flat below the slope of the hill.

The classic pinewood derby car is a great example of this phenomenon in practical applications:

EDIT: Solution fixed, thank you for the call-out.

I think this solution needs to be reviewed… By putting more mass further up the hill, you will build more potential energy which will convert to more kinetic energy.

This is different than a rolling cylinder which will roll at the same speed no matter the mass. This is because of the balance of rotational energy and translational energy with a rolling cylinder. A car has a bunch of mass that doesn’t rotate and only converts into translational energy.

Check out this video (pinewood derby) :