How not to shoot a monkey: video analysis of a classic physics problem.

I came across a neat video, via Jennifer Ouellette, where a couple of MIT students re-enact a classic physics textbook problem. It’s a problem that I first heard over a decade ago, when I was in high school, and is one of the few physics 101 problems to have earned the distinction of its own wikipedia page.

Here’s the setup. A monkey hangs from a branch of a tree. A hunter aims their rifle at a monkey. At the very instant the hunter pulls the trigger, the monkey gets startled by the sound, lets go of the branch, and falls from the tree. The question is: will the bullet still hit the monkey? If not, where should the hunter have aimed the gun to hit the monkey?

Source: UCLA physics lab manual

So, do you think the hunter should aim the gun:

  1. Above the monkey?
  2. At the monkey?
  3. Below the monkey?

Before reading on, take a moment to come up with your answer.

Thought about it?

This problem has somewhat of an amusing legacy. In an effort to revamp physics problems to fit more environmentally enlightened times, textbook authors have taken great pains to distance themselves from the barbaric act of shooting monkeys on trees.

Here’s the original version of the problem, from 1971, featuring a hunter and a monkey.

monkeyshot1

Shooting the monkey. Figure from Tipler, 1st Ed. (Worth, 1971)

Compare that to a modern variant, this one from 2000, featuring a distressed zookeeper who’s trying to coax an escaped monkey to climb down a tree. In the words of the authors, “After failing to entice the monkey down, the zoo keeper points her tranquilizer gun directly at the monkey and shoots.” If this is still a little alarming, some versions feature a friendly naturalist in place of the distressed zookeeper.

monkeyshot2

Sedating the monkey. Sears and Zemansky, 10th Ed. (Addison Wesley, 2000)

Here’s someone trying to feed a monkey a banana (I doubt the zookeeper would approve).

monkeyshot3

Feeding the monkey. Lea and Burke (Brooks/Cole, 1997)

By the time I came across this problem, it had become somewhat more convoluted. I mean, well.. just look at the figure.

monkeyshot4

Umm… where’s the monkey? Haliday, Resnick, Walker, 5th Ed. (Wiley, 1997)

I believe what we have here is someone blowing into a pea shooter that shoots out tiny spherical magnets, which can then stick to a falling metal can. The can is somehow wired to fall at the exact moment she launches the magnet. You know, just your every-day magnetic pea shooter wired to a falling-metal-can scenario.

And that isn’t even the strangest version of the problem I’ve come across. That honor goes to this next version. See if you can figure out what’s going on from the figure.

monkeyshot5

Giambattista, Richardson, Richardson (McGraw Hill, 2004)

This is, of course, the less famous cousin of William Tell, who decided to shoot a coconut with an arrow. Oh, and the coconut happens to be held by a monkey. Unfortunately, the monkey is a somewhat unreliable stooge, and the moment the archer releases the arrow, the monkey lets go of the coconut. Silly monkey, you had one job! Just hold the darn coconut.

Needless to say, these figures are starting to get a little visually jarring, and perhaps detracting from the key physics principle.

The latest version of this age-old conundrum comes to you from two MIT students, who wired a sock puppet monkey to fall at the exact moment a golf ball cannon is fired. I decided to track the motion of the ball and the monkey in the video. Before watching the video, think back to your prediction.

Isn’t that neat? Even though the golf ball curves away from its aimed trajectory, it still hits the monkey dead on!

So why did this happen? First, look at the light blue curve above. The monkey falls downwards in a straight line. But say you were to plot the height of the monkey, measured from the ground, as it changed over time. What would that plot look like? If you haven’t seen this before, it’s kind of surprising.

MB1

What you see is that even objects that fall in a straight line trace out a neat curve, called a parabola, when you plot their height versus time. The red curve is the monkey’s trajectory, recorded from the video, and the black line is a curve representing a perfect parabola. See how nicely they line up! Physics isn’t just textbook stuff.

Now, let’s add the height of the bullet into this picture:

MB2

Again, notice how well the bullet’s motion lines  up with a parabola. This is the sort of thing that I find very cool about physics – you can abstract away the monkey, and discover a mathematical world that’s hiding  beneath.

When I look at this curve above, it strikes me a pretty startling that those two curves intersect. It seems like a cosmic coincidence that the bullet managed to hit the monkey. But this isn’t the whole picture.

Let’s imagine for a moment what would happen in a world without gravity. The bullet would just keep moving in a straight line path. Let’s call this the aiming line. The monkey would still be up in the tree (since it can’t fall without gravity). It’s obviously going to be a bulls-eye shot.

Now, switch on gravity. The bullet curves away from its original, intended path (the aiming line, shown in green in the above video). And the monkey falls from its perch. But here’s the kicker: both the bullet and monkey deviate from their original paths at exactly the same rate. What I mean is this: if at any moment, you measure how far the bullet has dropped below the green line, and at that exact moment, you measure how far the monkey has fallen from its perch, those two distances will be exactly the same.

The bullet and the monkey both ‘missed’ the branch, but they missed it by exactly the same amount! If you think about it, this single fact means that they are still going to collide.

Let’s try it out and see if it works. Let’s measure how far the bullet strays from its original green aiming line. Here’s what this deviation looks like:

MB3

Surprisingly, it’s still a parabola, but a different parabola from before (in technical terms, we’ve subtracted off the linear term).

Now, we can do the same thing for the monkey. At zero seconds, the monkey sits on the perch. A tenth of a second later, it’s a few centimeters below the perch. Another tenth of a second and it falls further still. Let’s take this curve – the monkey’s deviation from its perch – and overlap it with the bullet’s deviation from the aiming line.

MB4

What do you know, it lines up pretty neatly.

This is why the bullet hits the monkey, why the archer hits the coconut, or why the magnet hits the tin can. It’s because the Earth affects the motion of all falling objects in exactly the same way.  No matter what you throw – coconuts, peas, golf balls, or bullets – they all deviate from their ‘aiming line’ at exactly the same rate. All falling objects play by exactly the same rules.

Footnotes:

In reality, a target rarely drops out of a tree the moment you fire a gun. In fact, gun manufacturers already take into account the fact that bullets fall. When you set the sight on a rifle, what you’re really doing is correcting for how far the bullet will fall by the time it hits its target.

The many variants of the hunter-monkey problem above are from the slides of an excellent talk by Eric Mazur where he emphasizes the importance of using simple, non-distracting figures.

Want to learn more about falling, and “the problem of the Moon”? Then definitely check out this superb Radiolab segment in their episode Escape, and another cool one on falling cats and why we fall.

 

 

10 Comments

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  • Daniel

    I feel like the second graph that with the bullet’s and monkey’s height over time is misleading. Just because the two lines intersect doesn’t mean that they hit each other. For example, if the gun was aimed above the head of the monkey, it stands to reason that the bullet would pass over the head of the monkey as both fell. But the two lines of their heights would still intersect momentarily before the bullet passes over the head of the monkey when the bullet and the monkey are at the same height. So saying that because the two lines intersect means that the two objects actually hit each others is misleading I believe. They occupy the same space in the vertial dimension but not necessarily in the horizontal dimension.

    • You’re quite right, I definitely could have phrased that part better. Thanks for pointing this out.

  • aed939

    We are assuming away reaction time and the speed of sound. Both of these factors will delay the monkey’s drop from the moment of the firing of the gun. If we include these corrections, you should aim above the monkey.

  • Simon

    Why did you not write anything about the speed of the bullet? Because the parabolas will only line up, when the bullet is flying at the same speed, as the monkey is falling, if I am not mistaken. Also the monkey is falling, so it’s accelerating, but the bullet’s speed is decreasing. So the power of the shot is very important in this scenario. So why didn’t you mention anything about it anywhere?

    • Actually, the speed of the bullet doesn’t matter. A slow bullet will hit the monkey lower (because it’s had more time to fall), and a fast bullet will hit the monkey higher (it’s had less time to fall).

      The key point is that the bullet is accelerating downwards at exactly the same rate as the monkey and this is true no matter how fast or slow the bullet is. So, it will always ‘miss’ the perch by exactly the amount that monkey has fallen.

      In other words, the last figure in the blog post (deviation from the aiming line) will look identical, no matter how fast the bullet is. The bullet and monkey have totally different horizontal speeds, yet these curves match up perfectly because they are determined only by the downward acceleration (g), which is constant.

  • allbuss84

    Trick question, aim right at the monkey adjusting only for the rate of fall relative to the original position. A bullet travels faster than the speed of sound and the monkey will not hear the gunshot until a fraction of a second after it was already struck by the bullet and falls out of the tree as a result.

    • Ha! That’s a clever solution. Never crossed my mind that bullets are faster than sound.

      I guess the question does implictly assume the speed of sound is much faster than the bullet, but that’s not really true for a bullet.

  • Lee

    I had an awesome high school science teacher who put together a simple, but elegant experiment to demonstrate this concept. Using glass tube and adjustable stands from the chemistry lab, he constructed a blow gun that used a small piece of blackboard chalk as a “bullet”. Across the room, he used a lab-built electromagnet to suspend a metal disk (the monkey). The tube was aimed directly at the target. The wires to the electromagnet extended back across the room and were taped to the end of the blow gun in such a way that they crossed at the muzzle and made contact, but the circuit would be broken by the chalk leaving the tube. The exact instant the chalk leaves the support of the tube and begins to fall (in a ballistic trajectory), the circuit breaks and the target also begins its descent. Students were invited to try it. No matter how fast the chalk left the tube, it never failed to hit the target.

    It made quite an impression on me. A small piece of chalk hitting a tiny *falling* target from 30 feet away? Over and over again? I couldn’t stop scratching my head until I figured out the physics involved.

    It would be even more impressive to set up a similar experiment in which two ballistic projectiles are guaranteed to collide. I’m thinking of two blow guns of equal length, aimed directly at each other head on. (Technically, it should work at any <90 degree angle as long as the angles are identical and the tubes are co-planar.) Instead of an electromagnet, use individual PVC pressurized air tanks with electrically controlled valves (sprinkler valves). A battery and a momentary switch should fire both projectiles simultaneously and they should collide midway regardless of velocity. Hitting a bullet with another bullet. Sounds like a nice weekend project for me…

    • Lee

      I just watched the full video – rather than just the video analysis clip – and I realized the MIT kids performed the experiment exactly as I described. Sorry for the gratuitous description.

  • Richard Shagam

    It was interesting to see how the problem has been modified over the years to make it more PC (and I don’t mean personal computer). However, the author of the blog implies that the problem only dates back to 1971. In fact, the problem is much older. I know I had this in high school physics, back in 1967. And it was an old problem then. It would be interesting to find out who actually originated the problem. I’ll bet on Lord Rayleigh