I have seen several videos similar to this.
Real? Fake? How many tries did this take? Let the analysis begin. Before I do any analysis, let me state that I think this is not fake. I do not know that for sure, just my first guess.
How would I tell if it is real or fake? This is tricky. I can’t really get a good trajectory of the ball to make some measurements on it because of the camera angle (next time people, make sure you set the camera up perpendicular to the plane of motion and far enough away to avoid perspective problems – thanks!) Really, the best I can do is to look at the time. How long does it take for the basketball to get to the goal? How hard would it have to be thrown to do this?
So, how high is the ball thrown from? Here is a shot of the statue plus its base.
Using this and Tracker Video (yes, you can use it to analyze images too), I get that the walk-way is at about 95 feet above the ground. If the basketball is 10 feet from the ground (as normal), then the vertical height for the ball to drop will be about 85 feet (26 meters).
How far is the goal horizontally? This one is a little tougher. First, let me start with this frame from the video.
Pay attention to the light post and that tree. Using those two, I think I have pinpointed the spot of the goal.
Here is another shot from another angle.
I need one other thing. Information about the basketball. According to Wikipedia, the NBA basketball has a circumference of 29.5 inches and weighs 22 ounces. (this is a radius of 0.12 m and a mass of 0.62 kg)
How long was the ball in the air? Again, I can use Tracker Video – even if I don’t know position data, I can still get time. From the time it left the throwers hand until the time it hit the goal was around 3.43 seconds. For an added bonus, I have a couple of frames where the guy was throwing the ball. Plotting the position for this time I get (I scaled it with the diameter of the ball being 24 cm):
This is a plot of the total magnitude of the velocity as a function of time while the ball is being thrown. From this, the ball was thrown with an initial speed of around 10 m/s (22 mph). That seems reasonable.
Do I need air resistance?
I suspect I do, but let me do a quick calculation. Suppose there were no air resistance. In this case, how fast would the ball be moving at the end of the throw? I will use work-energy and take the system as the ball plus Earth (so there will be gravitational potential energy). This gives:
If the ball is moving at 24 m/s, how much air resistance would there be? How would this compare to the gravitational force? I will use the following model for air resistance.
Where ρ is the density of air, A is the cross sectional area of the ball and C is the drag coefficient (about 0.5 for a sphere). So, at this speed the air resistance and the weight are:
Since the air resistance force for this mythical final speed is greater than the weight, air resistance is not negligible. This means the easiest thing to do is a numerical calculation (intro to numerical calculations) with python (here are some numerical calculation examples).
For my numerical calculation, I will have the ball start 26 meters above the target and moving at 10 m/s. Looking at the video clip, I am going to use an initial launch angle of 21 degrees. Let the calculation begin.
Here is what I got from vpython.
I added the ground and the tower for effect. Two important things from this simulation.
- Time: The time for this ball to make its motion is 3.01 seconds. That is good. It is not exactly the same as the value from the video, but my parameters could be off a little bit. This is close enough for me (for now).
- Distance: I looked at the map and the video again. I am fairly certain about where the goal post is. It should be about 40 meters from the starting location. However, when I ran the simulation, the ball only ended up 20 meters away. This is not good.
Ok, there is a problem with the distance, but maybe my starting parameters are wrong. What if the ball were thrown at 20 m/s? How far would it go? This is a simple thing to change. If I do that, the ball is in the air for 3.4 seconds and goes 37 meters. Could the ball be thrown that fast? Maybe. I admit that my initial value for the velocity was just based off of a few data points from tracker video. Also, there was a scaling problem and such. I don’t put too much weight on that speed.
There is some other data I can look at. Using the camera from the shooter location, I can see the ball as it enters the goal. This is far enough away and the ball is mostly moving downward that I can get an estimate of its final speed and compare that to the numerical calculation. Here is a plot of the y-position of the ball at the end.
This shows the ball moving at a fairly constant downward speed of 17 m/s. In the numerical calculation, at the end of the run the ball was moving horizontally at 5.5 m/s and downward at 17 m/s. That is pretty close.
I am going to say not fake – although I have been wrong before (remember the giant water slide jump?). My reasons are:
- I can model this motion and get a similar time (with air resistance)
- The ball should be in the approximate area with an initial speed somewhere around 20 m/s
- The final speed of the ball in the video and in the simulation mostly agree.
I am stopping here – but not forever. There is another very interesting question: how hard is this to do? Did these guys have to try like one million times or did they just get lucky (or are they that good?) I am saving this for another post.