Calculating Terminal Velocity for a Ping Pong Ball: A Simple Equation

[profaith]

hey how do you find terminal velocity? let say if you need to find the terminal veolcity of a ping pong ball? anyone has any ideas? what kind of experiment can i conduct?

 

[xanthym]

hey how do you find terminal velocity? let say if you need to find the terminal veolcity of a ping pong ball? anyone has any ideas? what kind of experiment can i conduct?

The terminal velocity of spheres like ping pong balls are fairly easy to compute and measure. The motion of such a sphere in still air at standard atmospheric temp & pressure will have a Drag Coefficient approx constant at $$C_{drag} = (0.44)$$. The force $$F_{drag}$$ due to aerodynamic drag ("air resistance") when the sphere falls thru air under those conditions is given by:

$$:(1): \ \ \ \ F_{drag} \ = \ C_{drag} \, \rho_{air} \, \pi \, D^{2} \, V^{2} /8$$

where "ρ_air" is the air density, "D" the sphere diameter, and "V" its fall velocity thru still air. The sphere will rapidly reach terminal velocity at the "V_T" such that Drag Force exactly balances gravitational force on the sphere, namely its weight "mg":

$$:(2): \ \ \ \ F_{drag} \ = \ m \, g$$

$$:(3) \ \ \ \ \Longrightarrow C_{drag} \, \rho_{air} \, \pi \, D^{2} \, V_{T}^{2} /8 \ = \ m \, g$$

You can easily solve for terminal velocity "V_T" in the above equation.

The above value can be checked experimentally by dropping the sphere from an elevated level (tall ladder might do) in still air. Time the fall and divide the distance fallen (e.g., height of the level) by the time interval to determine approx terminal velocity. You'll need to determine experimentally (& with calculations) the drop height needed to obtain reasonably accurate results.


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[SpaceTiger]

The above value can be checked experimentally by dropping the sphere from a tall ladder in still air. Time the fall and divide the distance fallen (e.g., height of the ladder) by the time interval to determine approx terminal velocity.

I presume that you're assuming $$V_T << \sqrt{2gh}$$ for that calculation, right? A web search gave a terminal velocity of about $$10~ m/s$$ for a ping-pong ball, so your ladder would have to be a good bit greater than 5 meters. Not crazy, but a bit of a stretch for everyday purposes.

 

[profaith]

wow this looks quite complicated. but thanks loads anyway! is there any other method/experiments tt we can use to determine the terminal velocity of the ping pong ball?

 

[minger]

It's really not that complicated of an equation. We gave you C, your coefficient of drag as 0.44. D the diameter can be easily measured or even just looked up online (simply just the diameter of a ping pong ball). Gravity is known, mass can be measured, and the density of air can be found in charts for given air temperatures. Then, just solve for V. It would be much easier than measuring experimentally, and given the margin for error in the experiment, would probably be more accurate.