# Can a rate be proportional to a shape?

This question may be a little vague, but it has a point. I woke up this morning with an idea.

Let's say I wanted to design a projectile that has a velocity proportional to its 'shape'. When the projectile is launched (from a catapult or a gun or whatever it is), the distance it travels will approach a limit. For example, if it's fired very quickly, it will decelerate quickly, and have zero velocity at some prespecified distance from the launch point. If it is fired slowly, it will decelerate slower, but still only travel the same distance.

The question would then be, what shape is the projectile to accomplish this? To answer that I could use a differential equation if I could somehow make the shape a function and that function proportional to the velocity of the projectile.

How could I do that?

Also, the projectile is static and not mechanical.

(Like I said, I woke up with this idea so it may not make any sense.)

My guess is that it's not possible. If the shape is static, the drag is a constant and so the distance travelled isn't.

Another idea I had was: what if I could 'edit' the viscosity of the air? If the air behaved more like a non-Newtonian fluid, then this would absolutely be possible. So I guess my new question is: is there such a thing as a non-Newtonian gas?

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It's not entirely clear to me what you're asking. Are you trying to find a projectile that goes a prescribed distance when fired with an arbitrary speed? (For example, you tell me that you want a projectile that goes 10 meters regardless of the speed you fire it at. I tell you to use a 1 m radius sphere with a mass of 1 g.) – Snowball Sep 20 '12 at 16:16