*Physics
208A.*

Last updated on 1 May, 1998

The parabolic arc chosen by your shooting can vary the percentage of success. When a shot is launched, the vector velocity and the position relative to the basket is fully determined [assuming no shot blocking activity]. The intitial condition determines the success or not of the shot.

Here's a couple of studies of the shots made during the Sonics, Timberwolves game 1.

A Gary Payton 3 -pointer.

Please examine the displacement-time and velocity time graphs.

Please ignore the blue curve and the scale on the figures. The Green parabolic y curve is the vertical and the linear brown x curve represents the horizontal motion. Note that as usual, the vertical motion has constant negative accelertion while the horizontal motion has zero accelertion.

The points that are displayed are distorted by the fact that the camera had motion during the course of the shot which changes the x,y coordinates. This is not corrected in this study. The camera angle also distorts the actual spatial coordinates.

To overcome these optical distortions, we can use the time information and the known fact that the 3 point line is 22' away from the basket [horizontally]. Payton shot from just outside the line.

Vox = 22'/1.2sec = 22/3.3 / 1.2 = 5.6 m/s.

Voy = g (fall time) = 9.8*(1.2-0.62)= 5.7m/s

Vo = sqrt(5.6^2+5.7^2) = 8m/s

The Vx and Vy are almost equal. This means that the shot approached the rim with an entry angle of 45 degrees. An examination of other shots show that this angle of approach is quite practiced very often. The launch angle is slightly higher thatn 45 degrees because the launch postion is below the height of the rim.

The entry angle of 45 degrees is optimal for two reasons.

The ball has the smallest velocity possible for approaching the rim from a given shot distance. If the shot hits the rim, the ball won't bounce away so far. This is the "soft" shot. To further enhance the possibility of entry in case the ball doesn't enter cleanly, the ball is given a backspin to make the ball bounce upwards thus enhancing the geometeric possibility of entry. To reach the basket from the 3 point line, the velocity of

The apparent size of the hoop is still quite large so that a clean entry is still quite possible. The ball has about 1/2 the diameter of the rim. When the ball approaches vertically, then the rim size is biggest and most tolerant of deviations from the center. At 45 degree entry, the rim has a width which is 2X the ball and the depth is still 1.4 x the diameter of the ball. If the entry angle is too steep, the apparent depth of the rim becomes equal to or less than the ball diameter and a clean entry is not possible.

There isn't a total agreement as to this point of view, the following is a study of the free throw of Peeler of the Wolves.

In this shot, Peeler has launched a lower trajectory shot. Time of flight was 0.95 seconds and the range of the shot was 5.8m.

Vox = 5.8/0.95 = 6.4 m/s

The fall time from the highest point was about .45 seconds so Voy = g*t= 4.4m/s

Ventry = 7.8m/s which almost has the same velocity as Payton's 3 point shot from a larger range.

The angle was theta = atan(4.4/6.4) = 35 degrees which still gives a clean entry possibility which did occur in this shot.

If the ball does not go in cleanly, the advantage of this shot is that the ball has a vertical velocity which is smaller than the 45 degree shot and if the shooter gives the ball a lot of back spin, the horizontal velocity can be reduced significantly if the ball strikes the rim. If this shot hit the rim, it would not bounce very far. We often see the ball snuggle around the rim and then fall in when these players shoot while our shots seem to bounce helplessly away. All the details like backspin count!!

Green parabolic y is the vertical displacement while brown x is the horizontal displacement. | Green y is the vertical velocity, Brown x is the horiz. velocity |

Good players have wonderful control of the launch velocity which to be successful from 3 point land must be controlled to better than 1%. They have good mechanics which will decouple the horizontal angle and the vertical angle and launch speed. To do this, the players will stroke the ball with the elbow and hand lined up in a vertical plane which is aimed at the basket. This controls the horizontal direction. The upper arm segment is usually aligned with the vertical launch angle so that when the hand completes its stroke to give the ball the proper speed, the vertical launch angle is already set.

The antithesis of the good form is to launch a shot with one's elbow sticking out to the side. This tends to couple both horizontal and vertical angles and the velocity so that it is VERY difficult to control the shot.

When a shot is taken while running, the shot difficulty is highly increased as the running velocity is added to the arm launch velocity. This is especially true if one were to take a shot while running transverse to the basket. These shots are usually taken only at close range where the launch velocity tolerance is larger. If you are running at the rim while taking a shot, the finger roll is often used because the running velocity already supplies the horizontal velocity. Stroking the ball will give too great a horizontal velocity for a good shot. Those wild and crazy Payton shots shouldn't be taken as models for almost anyone else.

Launching a shot while jumping vertically is not so bad as the vertical velocity is usually quite small at any time and the velocity is close to zero at the top of the jump when the shot is usually launched. In fact, since the shot is launched from a higher level, the rim is slightly higher so that the height helps enhance accuracy. A technique used by many good players is to take pick up the dribble and plant both feet to jump vetically. This gives a bigger height plus a good platform for an accurate shot. Dale Ellis takes the pass while planting both feet and jumping for his high accuracy 3 point shots.

You can further explore the relation of launch velocity and success in clean entry to the basket by using a monte carlo program which is available on this web. The game permits you to choose a height and range for the shot. You can then choose the central launch velocity and launch angle. The program will put in a 0.5% random variation into the launch velocity and 1% random variation into the launch direction. It's assumed that you launch the ball with the accurate horizontal angle. The game keeps track of the successes and gives you a scatter plot of the shots in angle-velocity space with the successes marked with green. Have fun.

To use this, download the program basketball.exe into your own Win 3.1 or Win 95 PC. Then execute by double clicking on the icon in explorer or file manager.

Peter Brancazio has written a rather good article on the physics of basketball which can be found in "The Physics of Sports", p 86. In particular, have a look at the graph Vo(angle of launch) to see that the region of largest success is around 45 degree entry angle.