Fly Me to the Moon … With Elephants!

There’s some weird and wonderful stuff on the internet. I recently ran across an animation showing a Saturn V rocket during liftoff, but with a small modification. Instead of shooting rocket propellent out the bottom, this one shoots elephants.

Why? you might ask. See, the Saturn V itself was a real beast. The workhorse of the Apollo program in the ’60s and ’70s, it’s the rocket that launched all the famous missions to the moon. It required massive quantities of fuel to get off the ground, and this clip shows in a nice, intuitive, bonkers way just how fast it consumed the stuff. Check it out!

(To be clear, these are conceptual elephants, not real ones. No one wants to see the words “shoot” and “elephant” in the same sentence. I’m imagining big gummy elephants of equivalent mass.)

Just for fun, let’s fact-check this clip to see if the rate of fuel usage shown is accurate. Yes, this would technically be rocket science—but the good kind.

How Do Rockets Work?

A rocket gets its motion by shooting stuff out the back end. There’s lots of complicated physics involved, but basically it all comes down to a change in momentum, where momentum is defined as the product of mass and velocity.

Let’s start with the simplest rocket in the history of rockets. It’s a low-friction cart with a ball launcher mounted on top. Watch what happens when the ball is shot out the back.

Video: Rhett Allain

Before it was launched, the metal ball was at rest and thus had a momentum of zero. After it was shot, it had some nonzero momentum. According to the momentum principle, a change in the momentum of an object means there is a force acting on it.

Illustration: Rhett Allain

I labeled the force as Fc-b, where the subscript indicates the force that the cart exerts on the ball. That tells us the change (Δ) in momentum for the ball (pb) per unit of time (t).

Now, here’s the whole secret to rockets: Forces always come in pairs! If you push on an object, it pushes back on you with the same force. In our case, if the cart exerts a force on the ball, the ball exerts an equal and opposite force back on the cart. That opposite force is called thrust. It means the momentum of the cart also changes—it gets pushed in the opposite direction.

I know, with a single ball the effect isn’t too impressive. But if the cart kept shooting balls, you could get a significant amount of thrust. How much? Well, the thrust force depends on the rate of change of momentum of the balls (or whatever else) you’re shooting.

So let’s take the equation above and—remembering that momentum = mass × velocity—replace Δpb in the top with Δ(mvb). That gives us an equation for thrust (below, look at the second term) in terms of the mass and velocity of the balls we’re shooting:

Illustration: Rhett Allain

Now let’s rearrange. It’s usual to group the time increment (Δt) with the change in velocity, because that gives us acceleration. But we can just as well group it with the change in mass: Δm/Δt (third term above). Now I can write the effective thrust force as a function of the time rate of mass depletion (rm).

There are two key values here. One is the speed of the balls (vb) and the other is the rate (rm) at which they’re being ejected, measured in kilograms per second. Knowing the weight of a ball, you could easily convert that to balls per second. So if we want to increase thrust, we can either (1) shoot each ball at a higher speed, or (2) increase the firing rate—more balls per second.

Oh, yes—things can get more complicated. For one thing, as you shoot stuff out of a rocket, the mass of the rocket decreases. But let’s keep it simple.

Saturn V Thrust

Now, using what we learned, let’s return to the Saturn V. The whole goal of this rocket is to produce enough thrust to lift off the ground and accelerate as it moves up. According to this useful Wikipedia page, the Saturn V produced a thrust of 35.1 million newtons.

That’s HUGE. For comparison, the jet engine on a Boeing 737 has a maximum takeoff thrust of about 120,000 newtons. You’d have to fire nearly 300 of them at once, pedal to the metal, to generate that much force. My little cart would have to shoot more than 800 million balls per second to match up.

Thrust can also be specified in pounds. That 35.1 million newtons would convert to roughly 7.9 million pounds of force. Not by accident, that’s somewhat more than the 6.5 million–pound weight of the fully loaded rocket. The “more” is what allows it to accelerate upward.

Now we can estimate the rate of fuel usage. That page I linked to above lists the total fuel for the first stage at 2.16 million kilograms, with a burn time of 168 seconds. That gives us an average mass rate of 12,900 kilograms per second.

We’re almost done! All that’s left is to convert from kilograms to elephants. There’s a neat trick to do this, which you can use in almost any situation.

In general, to change the units on a number, multiply it by a fraction that is equivalent to 1. So in our case, let’s say a bull elephant has a mass of 6 tons, or 5,000 kg. We can multiply our mass rate of fuel depletion by the fraction (1 elephant)/(5,000 kg), as shown below.

If you look just at the units in the expression below, you’ll see that we can cancel the “kg” on the top and the bottom and we end up with 12,900/5,000 elephants per second, or:

Illustration: Rhett Allain

That’s not all. We can also calculate the speed at which these elephants must be ejected. Using our number for thrust, along with the mass rate (in kg/s), I get an elephant ejection speed of 2,721 meters per second—about 6,000 miles per hour.

Video Analysis

So let’s check the film! I can use my favorite Tracker video analysis software to estimate the mass rate and ejection velocity in the animation. For the mass rate, I count about 6 elephants in 0.3 seconds, or 20 elephants per second. Hmm … that’s a lot higher than my 2.58 per second. The creator of this animation must be using smaller elephants. Either that or I miscounted. (It’s not easy to count ballistic elephants.)

What about the elephant speed? Here is a plot of the vertical position of one of the ejected elephants. Since this is vertical position vs. time, the slope of this line would be the vertical velocity (and therefore the ejected velocity).

Illustration: Rhett Allain

The slope coefficient on the fitted line is A. As you can see, it’s around 72 m/s. Oooh … that’s not nearly fast enough. Remember, we estimated an ejection velocity of 2,721 m/s. Which means that If you really did build an elephant rocket, it wouldn’t be this picturesque. The elephants would be just a gray blur as they whizzed past.

Bonus question: How do you think the velocity of the elephants (relative to the ground) will change as the rocket speeds up? It’s tricky. Got it? Answer: If they’re being shot at a constant velocity from a rocket that is accelerating away from Earth, the elephants’ speed relative to the ground would decrease.

In the end, this is a cool animation that illustrates how fast a Saturn V rocket uses fuel. It’s fun to walk through how you might create something like this. But it’s not a very realistic picture of the monstrous thrust force that an actual fake-elephant rocket would generate.

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