Ed note: Today marks exactly 10 years since Rhett first started writing physics posts for WIRED. Congratulate him on Twitter!
On the show The Boys, a speedboat smashes into a cetacean and the humans emerge unscathed. Could this happen in real life?
I think the best thing about The Boys on Amazon Prime is that it shows all the bad stuff that can happen when humans get superpowers. So it’s still a superhero show—but with a different viewpoint. That makes it interesting.
Of course I’m always on the lookout for physics problems to solve, and I found one in season 2 of the show. There is this scene in episode 3 in which the Deep tries to be a hero and stop the non-superheroes. Oh, the Deep is basically like Aquaman—he can swim underwater and he can talk to sea creatures (fish, yes, but also whales and other mammals). The Deep wants to stop a speedboat from escaping into a tunnel. To do this, he “asks” the whale to jump onto the beach and block the boat’s path. Oh, shouldn’t the beach stop the boat? Yeah, probably. I guess the Deep is just trying to overdo it, to show he’s a real hero.
In this scene, the boat does stop—but not before plowing straight into the side of the whale. The whale doesn’t survive, but everyone in the speedboat mostly walks away. Although it’s terrible that the whale died, at least we can get a nice physics problem out of this. (The whale was fake anyway.) In the show, we get a top-down view of this boat-whale collision as though it were filmed with a drone. This means that it’s perfect for a video analysis. Using software like Tracker Video Analysis, I can get the position of the boat and whale in each frame of the video. From this, I can find the velocity and acceleration of the boat and determine if real humans could survive.
Pretty much the first step in any video analysis is to determine the scale. In this case, I can use the length of the speedboat to set the distance scale. One small problem: I don’t actually know the size of the boat. I’m just going to guess it’s a 21-foot boat. That at least seems reasonable.
After setting the scale, now I can mark a point on the boat in each frame. Here is the data for the position of the boat as a function of time. This includes the time the boat is moving before it actually collides with the whale.
Here you can see that it appears the position of the boat can be described with three linear sections in the graph. You might recall from your introductory physics course that average velocity in one dimension is defined as the following.
This means that if the object has a constant velocity, then the ratio of its change in position (Δx) in a certain time interval (Δt) will be constant. This is also the formula for the slope of a line if I plot position vs. time—which I did. So, the slope of the linear fit for the first section of data will give the speed of the boat before the collision. From this, I get a value of 78 m/s, or 174 mph. Yes, that is crazy fast. Technically, there are some boats that can go that fast, but this is not one of those boats. That is just too darn fast.
Don’t worry, it’s probably not even a real boat and instead it’s most likely a computer-generated craft. So, why is it going that fast? Remember, the producers are trying to tell a story. Maybe they could animate it with a realistic speed, but maybe that doesn’t look cool. I’m completely fine with this change—and it gives me something to analyze.
So, let’s go with this initial boat velocity. Would the humans in the boat survive the collision? One of the key quantities to consider when looking at human survivability is the acceleration during impact. It’s the same as when your car accelerates and you get pushed back into your seat. Now imagine this multiplied by a factor a 10 and you can see how acceleration can be a killer. This is the acceleration caused by the boat going from its top speed to a slower value as it hits the whale—yes, that’s still an acceleration. It turns out that 22humans can handle an acceleration of about 200-300 m/s2 (20 to 30 g’s) for very short periods of time—at least according to experimental data. If the acceleration of the boat is under that value, maybe they could survive the whale impact.
In one dimension, acceleration can be defined as the change in velocity divided by the time it takes to change that velocity. It doesn’t matter if this change in speed makes the object go faster or slower—it’s still an acceleration. Some people would call slowing down “deceleration”—but physicists don’t like that word too much.
OK, here we have a small problem. From the video it looks like the boat goes from a velocity of 78 m/s to the next slower velocity of 20.4 m/s (from the slope of the second part of the graph above). But the acceleration doesn’t just depend on the change in velocity, it also depends on the time interval over which this change takes place. But I can’t really know the exact duration of the time interval. The data just looks like it changes instantly from one speed to the next. However, it’s a video where each frame is 1/24th of a second apart (0.042 seconds). Maybe during this frame transition, the boat changed speed. Using that as my time interval, the change in velocity from 78 to 20.4 m/s (45.6 mph) gives an acceleration of 1371 m/s2 (140 g’s). That’s not good.
In fact, if you want to have a “survivable” acceleration at around 200 m/s2 the boat would have to have the same change in velocity over a time interval of 0.29 seconds—that’s about 7 frames of video. Actually, if they changed this in the clip you probably wouldn’t even notice the difference. Maybe the humans did change velocity over a larger time interval. This would happen if they continued to move forward inside the boat as the boat slowed down. So maybe it’s possible. Also, I’m only looking at the change in velocity of the boat. If you want to consider the effect of this collision on the whale, that can be your homework.
Just for fun, I can again look at the acceleration of the boat as the motion transitions to the third constant velocity (the third slope in the above plot). This gives a velocity of 10.1 m/s (22.6 mph). Again, assuming a change in velocity over a time of 0.042 seconds, I get an acceleration of 245 m/s2. That’s more reasonable.
Yes, there’s a homework question to go with this problem. Suppose we have the boat with an initial velocity of vB1 (B is for boat) with a mass mB. It collides with the whale and becomes stuck inside, such that the whale and boat have the same final velocity (call it vB2)—yes, this would be an inelastic collision.
After the collision, the boat-whale combo slides a distance s across the sand before stopping. Assume that there is a constant frictional force on the whale (from the sand) with a coefficient of kinetic friction value of 0.8 (I just guessed at that). What is the minimum mass of the boat to get the whale to move a distance of 14 meters (that’s the value from the video). Yes, this is a more difficult problem—but you can do it. Here is a picture just to help you get everything organized.
Now for some hints:
Break this problem into three parts: Before the collision, right after the collision, and then once the whale-boat slides to a stop.
From right before to right after the collision, momentum is conserved. You can use this to get an expression of the velocity of the boat-whale after the collision.
Now consider the sliding part. There is a frictional force pushing on the sliding whale. Use that and the distance to get another expression for the velocity the whale-boat after the collision.
All that’s left is some math.
If you want to estimate the whale mass—I’m cool with that. The other option would be to solve for the boat-to-whale mass ratio.
If you want a bonus, see if you can model this scene in Python. That’s probably what I will do.
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