Should Spider-Man Swing or Run?
Clearly it looks cool when Spider-Man swings around on his webs. However, does this really save him time?
Once Spider-Man starts swinging, he becomes just like a pendulum. His motion is determined by the length of the web along with the starting angle. From a forces view, there is mostly just the tension in the web and the gravitational force pulling down on him. The key parameters to consider are the swinging angle, the time for one swing and the distance of one swing. Here’s a picture.
Before I get too far, let’s think about swinging as a form of transportation.
Tarzan Swing
First, there is the Tarzan way of swinging. Tarzan (you know, King of the Jungle) swings on vines instead of webs. He swings until one vine just about gets to it’s highest point and then he grabs another vine. Yes, the vine he grabs would probably be hanging vertically so that he would then start at the bottom of the next swing. This doesn’t seem very efficient so let’s just say he swings on vines and starts and stops at the same speed with the vines pre-set for him such that they are not vertical.
How do you model the motion of a person swinging on a rope like this? You might say “oh hey, it’s just like a pendulum. Pendulums are easy. They have a period of oscillation that depends on the length of the pendulum.” Well, that certainly is true. The time for one oscillation (there and back) is called the period of motion. Tarzan would just go half way (just there) such that he would have a swing time of:
But wait! This expression isn’t really correct. This is just an approximation for small swinging angles (the general rule of thumb is that it works for amplitudes less than 15° from the vertical). How do you get swinging times for larger angles? It’s not such a simple problem (but not impossible either). I like to solve this with a numerical calculation and a bit of python code. If you want to see how to do this, try this older post.
Once I have a working piece of code that can determine the time to swing, I can start to change things like the starting angle and the length of the web. Here is a plot of the average swinging speed for different cases. Oh, what do I mean by average swing speed? This is the distance from the start to the end of the swing (horizontally) divided by the time it takes to get there.
What does this all mean? Well, let’s first look at the 10 meter long web. If you increase the starting angle, you have a greater average speed. Why? Well, this also means that you start at a higher position so that as you move down you speed up more. For a large swing angle, you would be going very fast in the middle of the swing. But this doesn’t work for all angles. You can see that a starting angle of about 80° gives the best average speed for the swing since above that you waste time moving vertically both at the beginning and end of the swing.
What about the length of the web? Obviously, the longer the web the greater the average horizontal speed. Why? A long swing will sweep through a much greater distance in around the same time. Going farther in the same time gives a greater speed.
How about one more plot? What if I use a starting angle of 75° and just change the length? Here is a plot of the average horizontal speed as a function of web length.
So, even with a super-long web (60 meters) the average speed is still around 13m/s (around 30 mph). That’s pretty good, but I bet Spider-Man could run that fast. I mean, he’s Spider-Man.
Spider-Man Modified Swing
Why would Spider-Man swing like Tarzan? That is the way normal humans swing, but not Spider-Man. He can do better, right? Tarzan starts the swing at rest and then goes through the swing until the vine stops. What would be different about Spider-Man? He will swing and then before the swinging stops, he will let go. This means that he can “fly” through the air and shoot another web. Thus the Spider-Man swing will be a swing followed by a projectile-like motion.
Let me just pick some values and plot the trajectory of something like this.
In this swing, Spider-Man start with an initial speed of 8 m/s on a web that is 20 meters long and at an angle of 45°. The trajectory for this part is the blue stuff (he is moving to the left). Once he gets to the other side of the swing (but not the highest point since then he would be at rest), he lets go of the web and just becomes a projectile motion thing. The green part of the trajectory is the time that he is in the air returning to his release point where he would shoot another web and start all over.
Now I can just look at how far he traveled horizontally and the time this took to get his average horizontal velocity of 15.9 m/s.
This is much better than the maximum speed with a Tarzan-style swing of the same length (about 8 m/s). However, this is mostly due to his high starting speed of 8m/s. In fact if you just look at the part of his motion while one the web, he has an average horizontal speed of 14.3 m/s which is around the same value as the total.
But what’s the answer? Would Spider-Man be better off just running? I am going to say “no”. Why? Well, there is the obvious reason of looking cool while swinging. Also, just imagine Spidey running along the ground to get somewhere. People would try to stop him to say “hi” or get his autograph or something. That’s no way to travel. Also, I suspect that playing around with this Spidey-style swing, I could probably optimize it to get an even greater average speed. Maybe I will just leave that as a homework assignment for you.