Topic 4: Velocity

Standards

1. 8PC1.d. Students know the velocity of an object must be described by specifying both the direction and the speed of the object.
2. 8PC1.e. Students know changes in velocity may be due to changes in speed, direction, or both.

Classroom discussion.  Back to the basketball player making the jump shot. To get the ball in the basket the player has to do two things: aim the ball right (both up-and-down and left-to-right) and give it with just enough speed. Both are important, and if you know both you can predict whether the ball is going to go in as soon as it leaves the player's hands (unless an opposing player interferes with it on the way).
Investigations.  Give the students a simulated ball superimposed on a video of someone making a jump shot, as seen from the side. The student can control the starting velocity of the ball in 2 dimensions -- either by manipulating an arrow that represents the velocity 2-vector directly (polar coordinates) or by independently manipulating the x- and y-coordinates of the velocity (rectangular coordinates). The model has an appropriately scaled "force of gravity" built in. The challenge for the student is to match the velocity of the basketball in the video clip, and thereby have the simulated ball go through the simulated hoop.
Featured software

• Maze Game Use velocity vector.

Extensions: Model ball won't actually follow the trajectory of the real ball because of drag which basically acts to slow the ball down. For basketballs moving through air at speeds on the order of meters per second (Reynolds number ~100,000) the flow around the ball is largely turbulent and the air resistance is proportional to the square of the ball's speed. The coefficient of proportionality depends on a variety of factors, including most prominently the air temperature and the roughness of the ball's surface. The student is given control over this coefficient and the task is to vary it to simulate most accurately the actual trajectory. An alternative extension would be to find other initial velocities that result in the simulated ball falling through the hoop, and perhaps to map out the range of speeds and directions that "work," thereby getting a feel for just how accurate the basketball player must be to make the shot.
Suggested lab.  Shoot baskets and make videos and then use video capture software to measure velocity components. Alternatively, use acoustic or other velocity probes to measure velocity (caution: hard to do in two dimensions).
Assessments.  Use the same simulation software, but set up to solve a differnt problem -- for example, to measure horizontal range (the real-life situation could be a seiries of shot put trials). The challenge is to throw (put?) the shot as far as possible in three trials. Students can aim the shot initially in any direction and can give it any velocity up to some maximum, independent of direction. We monitor the input parameters (speed and direction) for each of the three trials, looking for: (1) do they always use the maximum speed available to them? (2) how close do they come to the optimal (45 degrees to horizontal) direction? (3) do they use feedback appropriately (in other words, do they come closer each time to the optimal angle)?