Sometimes I get requests. Sometimes they are requests for interesting things. Here is the video in question:

Could I “please, please, please analyze this?”

Of course the question comes up: is it real or fake? My first approximation says it is real. Clearly, this kind of thing is possible. However, I have been wrong before and this isn’t impossible to fake.

There are a couple of things to check. The first thing that came to my mind was the time of flight. I can get this from the video. If the time in the video is significantly different than what would be calculated, then it is likely fake.

Starting Assumptions

• No air resistance. This is wrong, but it is a good first approximation. Also, it will make it much easier to calculate.
• The reporter starts at half court. This makes the horizontal shot distance about 84 feet/2 = 42 feet and take away one foot for the goal off the back edge. (12.5 meters) Oh wait. According to this site, the center of the hoop is about 5 feet from the back edge. This would put the distance at 37 feet (11.3 meters).
• The initial starting position of the ball is about 6 feet (over the reporter’s head). The basketball goal is 10 feet off the ground. This gives a vertical change in height of 4 feet. (1.2 meters)
• The time of flight for the ball was right around 1.5 seconds (from the video).

Ok, there is one more thing I can get from the video. If I assume a basketball diameter of 29 cm, then I can get a few frames of the ball as it is being thrown. This is a plot of the vertical motion of the ball as it is being thrown.

So, I can use an initial launch speed of about 12 m/s.

Projectile Motion

Since I am ignoring air resistance, I can treat this as a plain old projectile motion problem. Here is a rough diagram.

The key to projectile motion is that you essentially have a y-motion and an x-motion. The only thing these motions have in common is the time it takes. Since there is no acceleration in the x-direction and the y-acceleration is -g, I can write: (oh, the ball starts at the origin for simplicity and I am calling x and y the final position of the ball)

I have two equations and two things I don’t know: time (t) and the launch angle (?). Now I will use the x-equation to solve for the angle and plug that into the y-equation. This is what I get:

Now, here is a trig trick. In y-equation, it has the sin of ? but I have cosine. Look at this fake triangle.

See, this triangle has the same value for cosine of ?. Also, I can use the Pythagorean theorem to find the other side. Now, what is the sine of ??

Substituting this into the y-equation, I get:

To solve for t, I will get the square root on one side and square both sides.

This is like the quadratic equation – well, for t² instead of t. I can use the quadratic formula to solve for t².

Now, I can put in the values from above to get the possible values for time (there will be more than one possible answer).

This gives two times of flight. 2.06 seconds or 1.12 seconds. What is the difference? Well, the longer time is for a higher angle and the shorter time is more of a “straight” shot. Neither of these values is exactly 1.5 seconds. Well, that might be ok. My method for finding the initial velocity was a little uncertain – especially with the poor quality video.

Fake or Not?

Unfortunately, I am going with “inconclusive”. Even if the time was exactly what I calculated, it wouldn’t absolutely mean it is a real video. I mean, if you can fake the video you could probably calculate the real time. Or better, just throw another ball and see how long it takes – even if it misses. So, I will just assume it is real for now.

What Are the Chances?

This is a better question. Clearly, it is possible to make this shot, but how unlikely is it? I will leave this as a homework question, but here is a tip. You will probably need to simulate this shot and run it a whole bunch of times. See the links below for an example.

See Also: