Race with Sticks - How to select the best one?

[czytron]

Hi!

How shall I correctly describe the velocity, as function of time, for an object put in a stream of e.g. water?

My friend wanted to challenge me in the game where a stick is thrown in the water and the winner is that who's stick first reaches a certain point. Ofcourse I picked a stick with low mass and large area to make the acceleration high, but how do I describe the situation physically?

Constant acceleration is no correct assumption since the force from the water, affecting the stick, will be smaller and smaller since it's relative velocity to the water decreases, while the mass remains the same.

What physical and mathematical approach is correct here to solve this problem?

/Happy for replies

 

[HallsofIvy]

Playing "Pooh sticks"? Any stick will move at the speed of the water it is floating in. As long as the stick does float, either area nor mass is relevant.

 

[jbriggs444]

A light weight stick will tend to float high in the water. This will expose it more to the wind and less to the water. A large stick may extend deeper in the water.

It seems clear that on a calm day in a deep stream, a stick with just barely positive buoyancy (so that it does not catch the wind) and good depth (so that it reaches water at depth that has not been slowed down by the air) would make better speed.

From experience boating, it also seems clear that picking the right launch point is far more critical than picking the right stick.

But the thrust of your problem seemed to be asking for an equation. The simplest equation would be one where the net force on the stick is directly proportional to its speed relative to the water. This leads to a differential equation of the form. dv/dt = -kv. This is one of the simplest types of differential equations there is (first order homogeneous linear). Its solution will be v = f(t) = [some multiple of] e^-kt.

The k will depend on the viscosity of water and the mass and shape of your stick. The multiple will be the one that makes your starting velocity come out right.

 

[czytron]

Thanx!
Yes ofcourse every stick's terminal velocity will be that of the water, so the thing I'm interested of is the way to terminal vel.,
The type of DE seems to make sense, just I was not really sure about the proportionality assumptions.

 

[CWatters]

Drag is proportional to velocity squared..

http://en.wikipedia.org/wiki/Drag_equation

The equation also contains the "reference area" which is typically the cross section presented to the flow... suggesting it's best to drop your stick at right angles to the flow.

 

[A.T.]

From experience boating, it also seems clear that picking the right launch point is far more critical than picking the right stick.

You can throw some floating particles (grass seeds) onto the water to see where it flows faster.

It also depends on how you drop the stick. If you hold it parallel to the flow and tilt it forward, it will convert some of the kinetic energy from the fall into forward motion during the dive. It might even temporally outrun the water.