Dot Physics

Basketball shot – real or fake?

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?

Gathering Data

Luckily, I know where this takes place – Vulcan Park in Birmingham Alabama. According to the Vulcan website, the statue is 56 feet tall and sits on a 124 ft pedestal.

So, how high is the ball thrown from? Here is a shot of the statue plus its base.

i-edd493e03f28e3f6fdcac648d46c6251-2010-08-06_vulcan_2_1.jpg

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.

i-6f27d7faf0f351eec33e2a6a79a70d5b-2010-08-06_frame1.jpg

Pay attention to the light post and that tree. Using those two, I think I have pinpointed the spot of the goal.

i-be8473a11a0cf98740a5ee205db2775d-2010-08-06_bing_maps_2.jpg

Here is another shot from another angle.

i-99c5b123f592c7d2619018641d1030d9-2010-08-06_bing_maps_1.jpg

Using this location, Bing Maps and Tracker Video the goal was probably about 130 feet (40 meters) horizontally from the throwing place.

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)

Time Measurement

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):

i-0e5f299f051ac47226bf7e2de34728bc-2010-08-07_data_tool.jpg

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:

i-5b06b80e471a3304c12e2e45c815d645-2010-08-07_la_te_xi_t_1_20.jpg

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.

i-1be028cbdae321344f6c25df36de42f8-2010-08-07_la_te_xi_t_1_21.jpg

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:

i-ab9b927229b846383fbaf46d25717924-2010-08-07_la_te_xi_t_1_22.jpg

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.

Numerical Calculation

Here is what I got from vpython.

i-3be8b3f8dbb7333cce81091de17c84fb-2010-08-07_bball.jpg

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.

i-c19c91f81ca1122a073222f7baa2e025-2010-08-07_data_tool_1.jpg

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.

Conclusion

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.

Next

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.

Comments

  1. #1 The Virtuosi
    August 7, 2010

    Nice post. But I think you meant to leave off the last line, seems like a note to yourself while you were writing.

  2. #2 Rhett Allain
    August 7, 2010

    @The Virtuosi,

    HA! You are right. It was a note to myself. I removed it. Thanks!

  3. #3 kleer001
    August 7, 2010

    Tee hee, you don’t have to fake it. Just show the 1 out of 1000 takes that made it.
    Is kinda like evolution, toss out what doesn’t work, keep what does.

  4. #4 Thomas
    August 7, 2010

    I suspect fake. I think, although this is just guessing, that the impact on the goal would o be a lot more violent than shown from that height. If you missed and hit the ring it would almost certainly break, and considering the number of tries you’d need to get that one, perfect hit, it would be an expensive video.

  5. #5 mark
    August 7, 2010

    thanks, I wonder what the margins of error are in the angle thrown or initial force.

    question:

    if the ball landed on a young squirrel would it kill it?

  6. #6 David
    August 7, 2010

    the ball hits the net 3 meters above the ground.

  7. #7 Scott Post
    August 7, 2010

    I am not sure how much difference it would make, but I have read that the drag coefficient of a basketball is actually 0.25. Like a golf ball, the dimples trip the transition to turbulence, reducing the wake effect. I have not seen experimental proof of this value of drag coefficient. It is on my list of things to test.

  8. #8 CCPhysicist
    August 8, 2010

    It really doesn’t look like he threw it that hard, but athletes can put a lot on a ball with little effort. I’ve seen a college bench player make a turnaround jumper from the sideline at half court.

    I would position the goal after throwing a few basketballs off of the tower. Then you are relying on the ability of an amateur athlete to repeat a physical action and only have to worry about windage.

    The ball is traveling close to vertical, which helps with the margin of error. The jumpers LeBron can make from the baseline (about 94 feet) are tougher.

    Impact of the ball on the rim is nothing compared to a dunk. These portable baskets are pretty tough.

    I’d suggest some model confirmation experiments, both real (drop a ball off of a tall building or take a road trip to that tower) and video (tracker analysis of LeBron’s baseline shots). You also have the apex of the ball’s flight (the form line on the elevator tower above the door) and the time to that point to calibrate your initial speed/angle estimate.

  9. #9 Rhett Allain
    August 8, 2010

    @Scott,

    Good point. Really, someone needs to measure the coefficient for a basketball – or maybe this has already been done? If the coefficient is much smaller, he wouldn’t have the throw the ball as hard.

  10. #10 Flavin
    August 8, 2010

    You mentioned your values for initial speed, but you didn’t give us the throw angle. Did you just simulate a horizontally thrown ball? I’d estimate the angle at 10 degrees or more above horizontal.

    Adjusting speed and angle separately could allow you to keep your fall time accurate while getting your horizontal distance closer to the theoretical value.

    (I should say that other commenters have mentioned the throw angle, but I want to address that explicitly.)

  11. #11 Flavin
    August 8, 2010

    Looking at the video clip, I am going to use an initial launch angle of 21 degrees.

    D’oh. Of course as soon as I post my comment I find where you answered my question. Reading skills. I need them.

  12. #12 CCPhysicist
    August 8, 2010

    @9:
    You can learn all you need to know by dropping it about 100 feet and analyzing the video to find the velocity dependence of the acceleration.

  13. #13 Nick
    August 9, 2010

    I know that genuinely skilled people can put a lot more into a throw than a neophyte, but 20 m/s is 45 mph. As I recall from an experience with a pitch radar at a ballpark one time, I can throw a baseball in the vicinity of that speed by putting my entire body into it. (I know that professionals can throw twice that fast, but they’re one-in-a-million freaks of nature (in the good way) who still, eventually, pull their own joints apart as often as not.) I’d characterize the video dude’s throw of that basketball, which is much heaver and more awkard than a baseball, as a “casual toss.”

    Let’s try a more quantitative approach. I’ve seen (real) videos of people sinking full-court shots. They look like they’re throwing the ball with a whole lot more effort than what video dude was putting out. A basketball court is aobut 55 m long. Assuming a player throws the ball up at a 45-degree angle (probably not optimal for making the basket, but let’s try it as a first guess), and neglecting air resistance (again, might be a problem), the player would need to throw the ball with a velocity of sqrt(((9.8 m/s^2)*(55 m))/4) = 11.6 m/s. At a (more realistic?) steeper trajectory of 60 degrees above horizontal, then the required velocity is sqrt(((9.8 m/s^2)*(55 m))/(sqrt(3)/2)) = 24.9 m/s. This is within the ranges of speeds shown in the video, but again, good basketball players look like they have to work pretty hard to put the ball up with these kinds of velocities, and video dude is not working that hard.

    Barring some revision to Rhett’s modelling, I’m going to guess that the video is fake.

  14. #14 Eric
    August 9, 2010

    OK, for all you skeptics out there, first required viewing before you debate me here> http://www.youtube.com/watch?v=lhbISRqSXW0. I pose this. Why would these dudes go through all that, then fake the swish shot? They shot from 2 angles and you can watch the ball leave the tower and connect with the net. I’m just a layperson and simple math tells me at what speed an object falls to earth and the time it takes to get there. The seconds from when the ball leaves the tower to the net seem correct.If you still think that the dudes faked the shot, then email or call fox6 news and have them recreate the shot and let fox6 film it.

  15. #15 rob
    August 10, 2010

    Rhett: to alleviate the non-perpendicular to motion camera angle, can’t you do some co-ordinate transformation to estimate what the projection of the true trajectory would look like from the camera angle?

  16. #16 Esteban
    August 10, 2010

    I think that mapping the position of the shadow of the ball in the ground may give some extra information about the trajectory to compare with the model, as a double check.

  17. #17 Rick Karle
    August 10, 2010

    I’m the sports anchor at the Fox TV station in Birmingham. The Vulcan statue is right next door to the station. We had two news photographers video tape these kids- it took them about 200 tries but yes, they made the shot, and no,the video is not doctored.

  18. #18 Rhett Allain
    August 10, 2010

    @rob

    I have tried something like this before with limited success. It is a pain.

  19. #19 Rhett Allain
    August 10, 2010

    @Esteban,

    Wow – I didn’t even think about the shadow. Great idea. I am not even sure how well it can be seen in the video, I will have to go back and take a look.

  20. #20 Rhett Allain
    August 10, 2010

    @Rick,

    200 tries? That is a very important piece of data – it will be useful later. Thanks!

    Did they get closer as the shots progressed?

  21. #21 Jill
    August 10, 2010

    The elevation from the base of the monument to the elevation of the goal was an additional 10 feet. I am the mom of the kid who is the founder of the Legendary Shots which made this Vulcan shot. Can you compare this the the Dude Perfect shot at the Texas A&M football stadium?

  22. #22 Rhett Allain
    August 11, 2010

    @Jill,

    I have seen the Texas A&M shot, and I wanted to analyze it. However, I found it was easier to get specifications of the distances with the Vulcan statue than the Texas A&M stadium. If someone has a few more details about that shot, I can probably put it together.

  23. #23 minero
    January 26, 2011

    all i have to say is that there is no way of that ball having a 20m/s – 72km/h initial speed but i think that it isnt fake.

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