Getting off the ground
For starters, we need some facts to work with. First, how far away is the moon? Since the moon's orbit around Earth isn't perfectly circular, there’s no one answer. But let's go with an average distance of 240,000 miles (386,000 km)—that's the number I think about when my car is getting old. Once I hit 240,000 on the odometer, I know I've gone far enough to reach the moon.
So you might think, OK, a human can pedal 15 miles per hour; I can use that to calculate the duration of the trip. Nope. You might be able to do that on a nice flat road, but in this case, you’d riding uphill—like, straight up. To add another complication, as you get farther and farther away from Earth, the gravitational field gradually weakens. Eventually, you’d be close enough to the moon that it would become a downhill ride and you could just coast.Instead of estimating speed, I'm going to estimate the power output of a human. If you are a Tour de France cyclist, you might be able to produce 200 watts for six hours a day. (Check out Ben King’s stage 4 ride on Strava.) I’ll use that value for now; you can change it later if you’re not an elite-level cyclist.
Now, how long would it take to move up a short distance Δy on your special moon-cable bike? Let's say the gravitational field has a strength g (in newtons per kilogram). The change in gravitational potential energy (Ug) for this short climb would be:
In this expression, m is the mass of the human (in kilograms). Since power (P) is the change in energy divided by the change in time, I can use my power estimate to find the time (Δt) it takes to move up a little bit:
Why am I using a short distance? It will be clear soon. First, a quick check: Suppose the human has a mass of 75 kg (165 pounds) and a power output of 200 watts. How long would it take to move up 1 meter? With those numbers, I get a time of 3.675 seconds.
Does that seem too long? Well, yes and no. Yes, it's true you could move up 1 meter of height on some stairs in, like, a second. But you’d be using way more than 200 watts of power. Imagine trying to keep up that pace for six hours straight. Yeah, so this expression looks good.
Dealing with changing gravity
Can we just do this same thing for the entire trip to the moon? No. The problem is that g factor. It might feel like gravity doesn't change as you climb up some stairs, but that’s just because you wimped out before you really got anywhere. The gravitational field weakens as the distance from Earth's center increases. We can find the (vector) value of the gravitational field with the following equation:
In this diagram, if you’re that gray dot out in space, we can calculate the gravitational force at that point using the equation on the right. G is a universal gravitational constant, ME is the mass of Earth, and r is a vector from Earth's center to you.
But wait! It's not just Earth that has gravity. The moon does too, so I need to add another term to my equation. Let’s say the moon has a mass of mm, and the distance from Earth to the moon is R. Now I can compute the total gravitational field:
I am sort of cheating by making the component of g due to Earth positive, but this way it will match the value on the surface of Earth from my previous calculation. Here’s a plot of the magnitude of this gravitational field going from Earth to the moon. (Here is the code.)
Starting on Earth, the gravitational field is 9.8 N/kg (and that's good). On the surface of the moon, the gravitational field is in the opposite direction with a magnitude of 1.6 N/kg. That checks out too: The moon’s gravitational field strength is about a sixth of that on Earth.
But for most of the trip, the effects of gravity aren’t zero, but they’re pretty small. Getting started would be the hard part, till you get up to about, oh, 10,000 miles, where Earth's gravitational pull is only 10 percent of what it is on the ground. That might seem far, but remember it’s 240,000 miles to the moon. And after that, you can really pick up speed. Finally, at the very end, it's an easy downhill to the lunar surface. Maybe a little too easy—more on that in a minute.
Your estimated time of arrival
Now that I have an expression for the gravitational field, I can repeat my calculation for travel time based on the human power output—this time recalculating g for each small step along the way. Here’s what I get for distance traveled as a function of time. It's not the whole trip, just up to the point where the ride switches to "downhill." (Here is the code.)
I'm actually surprised: It would only take 267 days. That's less than I figured! Taking our distance of 240,000 miles, that works out to an average speed of 37 mph. Of course, that is 267 days of pedaling 24/7 at a considerable level of exertion. If instead you pedaled for six hours a day, it would take four times as long—so that's almost three years, and it's not even all the way to the moon.
What about the rest of the trip? One option would be to just stop pedaling. You would mostly continue along at the same speed until you were much closer to the moon—but that's still pretty fast. Once you reached the surface of the moon, you would sort of crash. But how fast would this be? Here is a plot of bike speed as a function of time:
Yup. That's a fast moon bike—super fast. Sometime around day 258 you’d hit 100 meters per second, which is about 220 miles per hour. A week or so later you’d really be making good time, up to 1,000 m/s (2,200 mph).
When the gravitational field gets really small, all of the biker's energy just goes into increasing the speed. But really, there is an error in my model that would make it even faster (probably). My calculations consider all of the energy from the human going into gravitational potential energy to increase distance. But when the gravitational field is low, it really doesn't take much time to move "up"—so you end up super fast. This model doesn't directly take into account the changes in kinetic energy, and it assumes the rider starts with a zero velocity at the beginning of each step. But I still think the overall time calculation seems legit.
I guess it's a good thing the NASA astronauts used a rocket instead of a bike, though. Now for some homework.
- Where is the point at which the total gravitational field has a zero magnitude? This shouldn't be too difficult.
- In my calculation, I used a rider mass of 75 kg. That’s crazy small, as it doesn't include the mass of the bike. What if you change the total rider mass to 100 kg or maybe even 200 kg? How does that change the travel time?
- You can't ride that long without eating. Using a rider mass of 100 kg, how many sandwiches would need to be consumed to get to the moon?
- Since you can't just pull over at a Denny’s or something to eat, you'll have to bring the sandwiches with you. How much does that increase total mass?
- Why is there a cable running from Earth to the moon? Estimate the amount of steel needed to make a cable like this.
- The Earth-moon system is not stationary. Instead, it rotates. How would this rotation change the time needed to get to the moon on a bike?
- Come up with a plan for landing on the moon. How fast would you travel? When would you slow down? How much energy would need to be dissipated (in some form)?
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