# Coasting Down Hills

I live in a rural area. On my way home I go down a 1/2 mile long, moderately sloped hill. Most days I put my manual transmission car in neutral and coast down the hill. What puzzles me is that if I hit the top of the hill at 25 mph, the car will hit a max. speed of 44 mph at the bottom of the hill. If I slow to 2 mph at the top, the car will max. out at 42 mph at the bottom. Why is there so little difference in final speed?

Yours is the perfect illustration of the concept that resistance is not a linear relationship to velocity. It increases dramatically as speed increases, meaning it takes more energy to get the car from 25 to 44 than it does to get the car from 2 to 42. Since in this case the only energy you’re using is gravity, a constant, the times to get to higher speeds will be different.

To add my own illustration, if you started at the top of the hill doing 80 and coasting you might only reach 85. You might even hit the bottom of the hill still doing 80.

Well said…there are too many different variables at work, each a w/o linear relationship. Think of riding a bike. Whether you start at 0 mph or are "towed to 20 mph, your maximum speed is the same because of air resistance; and that’s just one limiting factor.

One way to think of it - if you drop a penny from the top of the Empire state building, it will be going the same speed when it hits the ground as if you threw it up, or down. In that case the wind resistance combined with the weight result in reaching terminal velocity long before it hits the ground, regardless of initial velocity.

Thanks for the feedback. So the resistance forces at work are: (1) Friction between the tires and the road surface; (b) Friction within the mechanical operations of the engine; and (3C) Air resistance.

Does the resistance in all three cases increase in a non-linear manner with velocity? It doesn’t seem that air resistance would be much of a factor in something with this much mass, but then I got a C+ in physics.

Air resistance is always a factor, regardless of mass, and will be a limiting factor…less drag by design may have higher speeds, but still there.

The wind resistance increases the most steeply (of a, b and c) with rising speed.
Doubling speed increases the force of wind wind resistance 4X.
Doubling speed increases the power to overcome wind resistance 8X.

I believe you are closing in on the terminal velocity for that hill.
“The terminal velocity of a falling body occurs during free fall when a falling body experiences zero acceleration. This is because of the retarding force known as air resistance. Air resistance exists because air molecules collide into a falling body creating an upward force opposite gravity. This upward force will eventually balance the falling body’s weight. It will continue to fall at constant velocity known as the terminal velocity.”

Also, when you hit the hill at 25 mph, you spend much less time on it than you do if you start at 2. Therefore, the accelerating power of gravity has less time to work. By my math, it’d take you 7 minutes, 51 seconds to descend the hill starting at 2 mph, while it would only take you 56 seconds to do it starting at 25 mph. Imagine pushing the accelerator (very lightly)for nearly 8 minutes rather than less than 1.

The math’s imperfect since it assumes a constant acceleration. It would, of course, taper off as mechanical and aerodynamic resistance increase, but the point still holds.

I had a 1.5 mile hill on the way to high school that would take my Accord from 45ish to 75 and let me coast the next 2 miles before I got back down below 40. I found out the hard way, though, that leaving a manual in neutral while driving is illegal in some states. (Such as PA.)