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If you throw a projectile from the ground at a certain angle, it's not hard to see that, assuming we're in a vacuum, throwing it at 45º from the ground will always make it go farthest before it hits the ground again. I wanted to find out what's the optimal angle if we take into account air resistance.

Before I go on, let me say that there's really no point to this question. I'm not trying to solve a practical problem. I just thought it would be fun to try to solve the equations.

This is the model I'll use: We'll assume that air resistance is roughly proportional to velocity (aka. Stokes drag; this fails for high speeds/Reynolds numbers, so let's not go there); in symbols, $$\mathbf{F} = -k\mathbf{v}.$$ We'll throw the projectile with mass $m$ from $(0,0)$ with initial speed $v_0$, forming an angle $\alpha$ with the ground, and let's say $\gamma = k/m$ because it will turn up frequently.

The differential equations are:

$$\begin{align} \ddot{x} + \gamma \dot{x} &= 0 \\ \ddot{y} + \gamma \dot{y} &= -g \end{align}$$

with initial conditions $(x(0), y(0)) = (0,0)$ and $(\dot{x}(0), \dot{y}(0) )= v_0(\cos \alpha, \sin \alpha)$. Solving (assuming I haven't made any mistakes), we get:

$$ \begin{align} x &= \frac{v_0}{\gamma}\cos \alpha (1- e^{-\gamma t}) \\ y &= (\frac{v_0}{\gamma} \sin \alpha + \frac{g}{\gamma^2})(1-e^{-\gamma t}) - \frac{g}{\gamma}t. \end{align} $$

Now the thing to do would be to solve $y(t) = 0$, substitute that into $x$, differentiate with respect to $\alpha$ and set it equal to $0$. However, that gets messy quickly, because the solution involves the Lambert-W function and it's all a mess. I haven't even tried to substitute into $x(t)$.

So finally, my question is: is there a simpler or numerical way to solve this? Is there even a single angle which will always work, or does it depend on the conditions? Can we find that out without actually solving?

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Mathematica is a good choice… They also have packages about this, –  Marvin Jan 26 '13 at 2:00
The air resistance is proportional to $v^2$ for all but the lowest velocities, so your equation isn't a good model for the real world. There is no closed form solution for the general case, though for vertical motion the equation can be solved. See –  John Rennie Jan 26 '13 at 10:07

3 Answers 3

up vote 5 down vote accepted

This is one of those problems that take on a much neater solution if you go dimensionless...

If you take a new time, $\tau$, horizontal coordinate, $\xi$, and vertical coordinate, $\eta$, such that

$$\tau = t \gamma,\ \xi = x \frac{\gamma}{v_0 \cos \alpha},\ \eta = y \frac{\gamma}{v_0 \sin \alpha}$$

you can rewrite your equations as

$$\ddot{\xi} + \dot{\xi} = 0$$ $$\ddot{\eta} + \dot{\eta} = -\frac{g}{\gamma v_0 \sin\alpha} = -\lambda$$

with initial conditions at $t=0$

$$\xi = \eta = 0,\ \dot{\xi} = \dot{\eta} = 1$$

This has solutions

$$\xi = 1 -e^{-\tau},\ \eta = (1+\lambda)(1 -e^{-\tau}) - \lambda \tau$$

and setting $\eta = 0$, we can get rid of $\tau$, and the range will be $\xi^*$ that fulfills

$$-\frac{1+\lambda}{\lambda}\xi^* = \log (1-\xi^*)$$

The maximum range will happen when $d\xi^*/d\alpha = 0$, so it is easy to show that at the maximum range:

$$\frac{d}{d\alpha}\frac{1}{\lambda} = \frac{1}{\lambda \tan \alpha}$$ $$\frac{d\xi^*}{d\alpha} = \xi^* \tan \alpha$$

Differentiating the equation for $\xi^*$ w.r.t. $\alpha$ using these two last formulas will, after a lot of cancelling and rearranging, take you to

$$\xi^* = \frac{1}{1+\lambda \sin^2 \alpha}$$

You can use this relation in the first expression we got for $\xi^*$ to get rid of $\xi^*$ and get a trascendental equation for $\alpha$ that looks something like:

$$\frac{\sin \alpha + \lambda^*}{\lambda^*}\frac{1/\lambda^*}{1/\lambda^* + \sin \alpha} = \log(1/\lambda^* +\sin\alpha) - \log(\sin \alpha)$$

where $\lambda^* = \lambda \sin \alpha$ and is independent of $\alpha$. This last equation you would want to solve numerically to get $\alpha^*$, the angle producing the maximum range.

While messy, the solution clearly depends on $\lambda^*$, so there will not be a single angle that always works.

EDIT Just run this last equation through a numerical simulation, and here's the optimal angle $\alpha^*$ (actually $\sin\alpha^*$) as a function of $\lambda^*$, showing how it grows ever closer to $\sqrt{2}/2$ for $\lambda^* \to \infty$.

enter image description here

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Are you used Buckingham π theorem for making dimensionless coordinates? –  m0nhawk Jan 26 '13 at 9:24
@m0nhawk Not really. I just chose the obvious units for velocity and time, and then $\lambda$ forms on its own. But you could get it that way too. –  Jaime Jan 27 '13 at 9:20

I feel tempted to give a bit of a twist to this question. As John Rennie mentions in a comment above, unless you are talking about slowly moving microscopic projectiles, drag forces are best modeled as being proportional to the square of the velocity.

This brings me to a human-interest story relevant to the problem under discussion.

Last summer, it was reported that a 16 year old Indian schoolboy had solved a 350 year old problem that had stumped researchers since the time of Newton. All of this was highly over-hyped. The story, however, is relevant to the present discussion as one of the two problems solved by the boy is that of the movement of a particle under a uniform gravitational acceleration and a quadratic deceleration due to air resistance. You can find a Physics SE discussion here, and comments from professors at Dresden University (where the schoolboy did an internship) can be found here.

the 16 year old schoolboy, Shouryya Ray, with a poster highlighting a constant of motion for a ballistic particle undergoing a quadratic drag force

As OP is "seeking fun in solving equations", it might be worthwhile to check out some of the above links.

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I built a high pressure (2000 PSI) pneumatic cannon that can shoot a 14 Oz finned aluminum projectile over 5 miles. After numerous tests the optimum angle proved to be 38 degrees. I roughly extrapolated a drag coefficient by comparing the measured distances with the so called ideal vacuum conditions, but it still does not explain the reason for the reduction from the ideal 45 degrees.

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Hi Edward! It is good that you have used experimental results to back your opinion. However, try to give a mathematical approach suitable for the problem to be solved, as per the question. –  Aniket Sep 26 at 9:14
This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post - you can always comment on your own posts, and once you have sufficient reputation you will be able to comment on any post. –  Jon Custer Sep 26 at 13:08
I second @Aniket and the suggestion that you match the mathematics to your experiment. Why do you say that the drag coefficient doesn't explain the reduction from 45 degrees? It absolutely does, if the math is done properly. –  Bill N Sep 26 at 15:40
Always nice to see experimental evidence show up, but this evidence is not related to this question, because a projectile that will fly five miles isn't in a linear drag regime at all. –  dmckee Sep 27 at 0:00
Aniket I think you misinterpreted Bill's comment. "I second @Aniket and the suggestion that you match the mathematics to your experiment." Means he is agreeing with you. –  dmckee Sep 27 at 0:13

protected by Qmechanic Sep 26 at 7:31

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