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Imagine I throw a ball straight upwards with some velocity $v_1$, and filming the ball with a camera, I can estimate a velocity $v_2$ (along the same vector) after the ball has moved a distance $D$. Using the difference between $v_1$ and $v_2$, and assuming constant friction due to air, how well can I estimate the initial velocity necessary to toss the ball some height $H$?

For fun - provided some $v_1$, $v_2$, and $D$, can we estimate an upper-bound for the Earth escape velocity with air friction/drag? Or is there unpredictable scaling of friction with velocity?

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2 Answers 2

In general there is no simple equation for the trajectory if you include the effects of air resistance, and you need to use numerical methods. However for the case where you throw the ball straight up this can be solved analytically. See the Hyperphysics article for the gory details. We're not supposed to just post links, but the solution is a bit messy and I'm not sure what would be gained by just duplicating the article here.

For all but very low speeds the air resistance varies as $v^2$. I gave a rough estimate of when the $v^2$ dependance ceases to be a good approximation in Limitations of drag equation. For a tennis ball the $v^2$ approximation ceases to be good at speeds below about 0.2 m/sec so you can probably neglect it.

There are a few problems with calculating escape velocity including air resistance. Firstly the air density, and therefore the air resistance, varies with altitude and this variation isn't given by any simple equation. Secondly when calculating air resistance you use a drag coefficient that is assumed constant for any particular shape e.g. 0.47 for a sphere. However the drag coefficient is only approximate and does vary at high speeds and especially supersonic speeds. Since escape velocity would be well above the speed of sound you'd need to account for the change in the drag coefficient. Given these two problems I'm afraid calculating escape velocity would mean going back to your numerical methods.

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I'm happy that someone had noticed Hyperphysics. Its an amazing website. I've read many articles in it. Though its articles are not so fancy in design, they're still well precise and provide a bunch of experimental data. Still, I like those :-) –  Waffle's Crazy Peanut Jan 12 '13 at 9:44

In practical terms camera measurement may be very inaccurate. The velocity change may poorly depend on drag. It depends on which sort of ball you use.

To model the situation I use the following ode (free fall with friction equation for unit mass in SI units):

$$ \ddot{x} = -9.8 - k\dot{x} $$ with $x(0)=0, x'(0)=10$

The first term is supposed to be earth's acceleration. The I look for a "time of return" $T$ and check value of $x'(T)$. I find that as soon as the value of $k$ exceeds 5, the dependence of terminal velocity on $k$ is relatively weak (and the measurement is inaccurate). Measuring with camera forces you to use relatively large speeds (try using 5 consecutive frames and using some sort of interpolation, 5 point stencil, high order differential scheme - this should help SIGNIFICANTLY with accuracy) which brings you closer to saturation of terminal velocity.

In practical term the answer to feasibility question depends on $k/m$ ratio and I think it is doable provided you do not let your ball close to terminal in a free fall (I would throw the ball not too high and use high fps camera - such as new Nikons... the capture 1200 fps).

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