This awesome video shows a shark shooting out of the water like a Poseidon missile launched from an underwater submarine (seen at the 1:37 mark). Normal humans think this is cool, but what does a physicist see? We see a projectile motion problem. Yes, for any object that has a motion that is due to the gravitational force only physicists call that projectile motion. Great, but what can we do?
If there was something next to the shark, you could do a comparison of size to get the total shark length. However, in this case the shark is all by itself. This is where – video analysis comes to the rescue. By looking at the location of the shark in each frame, I can get a plot of the vertical shark motion. But first, I need something to scale the video. Here I will set the coordinate axis and set the shark length to "1 shark."
Now I can estimate the center of mass of the shark to get the following plot of vertical position (remember the units for the distance axis is in "sharks").
Since projectile motion has a constant vertical acceleration of -9.8 m/s2, the position vs. time plot should be a parabola. By fitting a quadratic equation to the data, I can compare it to the following kinematic equation:
So the term in front of t2 should be (1/2)g. This means that I can make the following equation.
A 2.33 meter long shark would be 7.5 feet long. Clearly great white sharks can get bigger than that, but it seems like a fair estimate.
Now that I have the distance conversion of 1 shark = 2.33 meters, I can get the jump height. From the video, the shark's nose moved up 1.07 sharks or 2.49 meters (8.2 feet). As the shark moved up, it also turned. This means that the center of mass for the shark didn't go quite as high, only 0.66 sharks or 1.54 meters (5 feet).
Here is a classic introductory physics problem. A ball is tossed straight up such that it reaches a maximum height of 0.66 meters. How fast was it thrown? Now replace "ball" with "shark" — instant transformation from boring physics problem to great blog post. See. That's not difficult.
There are two methods I could use to find the "launch velocity." The simplest way is to use the time interval from when the shark leaves the water up to the highest point. The second method uses the maximum height. This second method is a little more complicated so let's use the time it takes to get to the highest point (0.8 seconds from the video).
With the time interval, I can use the definition of acceleration.
It's straight-forward to solve for the initial velocity since the final velocity is zero. Putting in a value for the time and g, I get a launch velocity of 7.84 m/s (17.5 mph). Is that fast? I think that's pretty fast.
Now for some problems for you.
- What is the mass of this shark? There are a couple of ways you could get this, but here's what I would do. I would approximate the shark as a cylinder with a density of water (1000 kg/m3). Make the cylinder a little bit shorter than the shark to account for the tapered shape of the actual shark.
- How much energy does it take to jump like this? You calculate this just by finding the initial kinetic energy (using the mass from the previous question).
- How many fish would a shark have to eat to get the same amount of energy as the jump? Is the jump worth it if the shark misses a fish (or seal)? I assume the shark does this move to catch food.
- Estimate the drag force on a shark of this size. Yes, this is tough but you can do it. With this drag force, what is the power needed to swim at a constant speed of 17.5 mph?
Ok, there is one more homework question. Find some way to incorporate "jump the shark" into this post. Yes, I know it is THE SHARK that is jumping — that's what makes this question rather difficult.
