terminal velocity again

[whisperblade]

okay so i posted earlier (while back) regarding terminal velocity. I have a parachute experiment set up, but i cannot find a way to calculate terminal velocity. does anyone out there have a formula i can use to find terminal velocity if all i got is a good old fashioned pencil, paper and stopwatch? any help would be appreciated.

[HallsofIvy]

The whole point is that the resisting force due to air depends upon downward speed while the downward force, gravity, is a constant. The simplest case, applicable to parachutes, for example, is when the resisting force is proportional to speed: Fr= -kv (I'm taking positive downward- resisting force for a falling object is upward) so the total force is F= mg- kv. The body will stop accelerating downward and move with constant speed ("terminal velocity") when F= mg- kv= 0 or v= mg/k.
Of course, what k is depends on the situation: size of parachute, etc.

For small, dense, objects falling through air (or even more, water), resisting force tends to be proportional to v2. The total force is F= mg- kv2= 0 so terminal velocity is v= √(mg/k).

[whisperblade]

so let me make sure i have this down correctly. so to find terminal velocity i take the mass of the object, multiply that by the force of gravity and divide that whole quantity by the surface area of the parachute?

thanks for the help by the way.

[ceptimus]

You need some extra factors in your calculation. There is the coefficient of drag for the parachute - this is just a number that refers to the object's shape. More streamlined objects with the same cross sectional area have a lower coefficient of drag.

You also need some constants to do with the density and the viscosity of the fluid (air) the object is falling through. A parachute's terminal velocity actually decreases a bit as it gets nearer the ground, because the air is thicker lower down.

These factors together make up the 'k' in HallsofIvy's formula.

[whisperblade]

see the thing that is hard for me is finding those things to make up k using only what i have for my experiment, which is like a stop watch and ruler stick thing and a video camera maybe. my teacher was saying how i need to find an easier way to find terminal velocity without doing too complex of calculations. which is where i'm stuck at right now.

[ceptimus]

OK. How about this.

Make a small parachute, say from a handkerchief or similar, and drop it from high building. Use the video camera to film the parachute dropping (you need a friend to help with this, of course, as you can't be in two places at once).

If the parachute stays reasonably close to the building (but not touching), and the camera is reasonably far away (the farther the better, as long as you can still see the parachute) then you have a scale (the windows on each floor of the building) and a timer (the frames of the film).

You can watch the film in freeze frame mode, stepping a frame at a time, and you might get something like this as the parachute falls.


floors frames frames
fallen of video for this
past (total) floor

1 40 40
2 70 30
3 95 25
4 121 26
5 145 24
6 170 25
7 195 25

You can see that after the first two floors, in this example, the parachute is falling at a pretty constant speed (25 frames per floor) so this is its terminal velocity. Now you need to know how many frames a second the camera makes, which you can look up, or measure by filming a clock, and how high each floor of the building is, which you can measure directly by lowering a tape measure from a window.

You can repeat the experiment with table tennis balls, tennis balls and so on and try to measure the terminal velocity for each one. Try to choose a day that isn't too windy for the experiment, as this will reduce errors.

[whisperblade]

thanks a bunch! helps tremendously!