Check out this awesome stunt by young Spider-Man star Tom Holland. Yes, this is for the new movie Spider-Man: Far From Home, but I really don't think it's a spoiler unless you weren't expecting Spider-Man to jump around.
Holland is a former gymnast and dancer (he played Billy Elliot onstage in London as a kid), so he has the chops to do a lot of his own stunts. But as you can see in this rehearsal clip, he still needs a harness to jump like a superhero.
Even though these are fake jumps, we can still use some real physics to analyze them and see how closely the on-film result would resemble a real superhero’s motion. Then, even more cool, we can use what we learn to modify the footage in a way that makes it more convincing.
Ready? Let's start with a short refresher on flying objects.
What Goes Up ...
When you toss a ball into the air, once that ball leaves your hand, there is only one force acting on it, and that’s gravity. It’s exactly the same if you’re the object, and you’re trying to execute a dicey parkour jump. Once you've taken off, the only force affecting you is gravity.
People get confused about this. You might think there’s some force pushing you forward through the air. Nope, that’s just inertia. According to Newton’s first law, if you shoot a clown out of a cannon, he will keep moving at the same speed in the same direction indefinitely—no booster rocket required. He only stops because gravity sucks him back to the ground. (OK, I’m ignoring air resistance, but let’s say this is a skinny, aerodynamic clown, so that's a relatively small effect.)
So first let’s figure out what the gravitational force is. That’s pretty simple. In general, it depends on the mass of the object in question (m) and the mass of the planet it’s on. We’ll stick to Earth, so that part is a constant. Then we can compute the force of gravity using this equation:
Here, g is the gravitational field, with a value of about 9.8 newtons per kilogram. If you’ve ever taken a physics class, you’ve heard that number so many times you still mutter it in your sleep. The gravitational field weakens as you move away from Earth’s surface, but so long as you don’t jump 100 miles into the sky, we can treat it as a constant.
So what does a force do to an object? It makes it accelerate. For gravity, that means a falling object will pick up speed as it plummets to Earth. (This is why it’s unwise to jump off tall buildings.) To be specific, it accelerates downward at a rate of 9.8 meters per second per second (m/s2).
Instead of jumping down, what if you jump up and forward onto a platform? The physics don’t care. You launch with a certain velocity, which we can break into horizontal and vertical components. The horizontal velocity, per Newton, never changes. But gravity immediately starts winding down the vertical part: Your rise slows until it becomes negative—i.e., downward—velocity, and then you fall faster and faster until the platform (or the ground) stops you.
And here’s the key point: It doesn’t matter if you’re an ordinary mortal or a superhero; the vertical acceleration is the same –9.8 m/s2. With superhuman leg strength, you get a better push-off, so your initial velocity is higher and it takes longer for gravity to turn you around. That means you can jump higher. But the effect of gravity should look the same.
Now, if we put this vertical acceleration together with the fixed horizontal velocity, we get something special, called projectile motion. You know it as the beautiful arching, parabolic path of a ball tossed across a room, or a cup of coffee knocked off a table, or any hapless thing launched into the air.
This is great, because it means that even the real Spider-Man would move in a way governed by introductory physics. So now I can measure Tom Holland’s vertical acceleration and compare it with what it would be without a harness—that is, if it were true projectile motion. That basically determines how convincing the illusion looks.
Supermen Are From Mars?
First, I’m going to plot the actor's spatial coordinates in each frame of the video using the free and awesome Tracker video analysis tool. Then I’ll try to fit a quadratic equation to the data. Why quadratic, you ask? Because if you graph a quadratic equation, you get a parabola!
For an object with constant vertical acceleration, the following formula describes its height (y) as a function of time (t). We call this a kinematic equation. y0 is the starting height, vy0 is the initial velocity, and a is the acceleration.
So now let’s look at Holland's first two jumps, the initial big one followed by a shorter one. This graph shows his elevation (in meters) on the vertical axis as time elapses (in seconds) on the horizontal axis. Oh, I scaled the video based on the actor’s height (1.73 meters, or 5'8").
You can see that Tracker fit a parabolic equation to the data on the first jump. This is what it gives:
The first thing to notice is that the equation (represented by the red line in the plot) fits pretty well. That’s great—it means Holland’s jump resembles a real, unassisted jump.
Now let’s look at the vertical acceleration. The term we want is the first one in the fitted equation, the t2 term. Our coefficient of –2.096 is equal to (½ a) from the kinematic equation, so the implied vertical acceleration, a, would be –4.192 m/s2.
Whoops. That is not the acceleration of a projectile on the surface of Earth; it should be –9.8 m/s2. This is basically what a Spider-Man jump would look like on a much smaller planet. Now that you know that, go look at the video again—you'll see there's a weird floaty quality to that first jump.
Fixing the Physics
You see this floaty effect in action movies all the time, so audiences are probably used to it. But the cold reality is that, given Holland's initial velocity, a superhero on Earth would have come down faster and slammed into the side of the platform. To stick the landing, he would have had to launch with way more speed . (By the way, that’s why baseball analysts are obsessed with hitters’ “exit velocity.”)
On the next jump, I get a vertical acceleration of –6.074, which is closer, but you can see that the fit is pretty janky. This whole video has an issue with repeating frames—successive frames that are identical—probably caused by someone compressing the file at some point. That messes up the analysis a bit, and since this jump isn’t as long, it’s harder to fit.
The other big jump is the last one in the sequence, where Holland has to go from a low platform to a high one. This was clearly a bigger challenge for the cable operator, and you can see a total failure of physics on the upward bound—he gets a sudden upward boost, as if gravity flipped into reverse. Well, what can you do? After all, he isn’t really Spider-Man.
But wait! Thanks to our knowledge of projectile motion, we can do a little postproduction magic on this clip to fix it. If the video ran faster, Spider-Man would appear to move up and down in less time, giving us a higher acceleration. With just the right frame rate, I can tune his acceleration to exactly –9.8 m/s2.
Let’s assume that the frame rate in the posted video clip was actually slowed down by an evil mastermind. In fact, I’ll say it has time units of “fs” (fake seconds). That means the acceleration on the first jump is around –4 m/fs2. If I set this equal to our desired acceleration in real seconds, I can solve for the conversion factor between real and fake seconds.
That means 1 second is equal to 1.57 fake seconds. That’s the ratio I need to speed up the video. (If I wanted to really polish the whole shot, I might have to use a different factor for each jump, but I’ll keep it simple and use this factor from the first jump for the entire clip.)
That's not perfect, but it does look more realistic. If you actually saw a superhero in real life, this is how fast they’d have to move to accomplish those jumps. Whether it makes for better cinema, I’m not sure. It’s kind of hard to appreciate the stunts when they fly by at that speed!
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