Sports: Basketball in Hoop!

The Basketball Free Throw Challenge 🏀

A basketball player is practicing free throws, trying to perfect their shot. The height of the ball in meters at any given time, t, seconds after release can be modeled by the equation:

where:

• v is the initial velocity of the ball (in m/s),
  
  

• h0 is the height of release (in meters), and
  
  

• h(t) is the height of the ball at time t.
  
  

The basket is 3.05 meters high, and the player releases the ball at 2 meters from the ground.

Open-Ended Inquiry Questions:

1. Understanding the Shot: If a player releases the ball with an initial velocity of 7 m/s, how long does it take for the ball to reach its highest point? What is the maximum height?

2. Optimizing the Perfect Free Throw: To increase the chances of making the shot, the ball should reach the basket at the highest possible arc. What should the player’s initial velocity be to ensure that the ball reaches exactly 3.05 meters at its peak? 

3. What is one weakness of the answer you calculated in #2 above? (hint: what piece(s) of information was/were missing or assumed in your answer? (what assumptions or simplifications were made in your answer calculations)) 

4. Sports Science Extension: If the player wants to find the best shooting angle that maximizes their margin for error (meaning the shot reaches 3.05 meters with a gentle downward motion), how might we model this scenario using quadratics and optimization? You decide that the best shot should have the vertex (max. height) of the parabolic arc reach 3.2 m midway between the player and the basket. If the player stands 4.5 m away from the basket, what should be the release angle to make this happen?

5. Adjusting for Real-World Factors: If the player adjusts their release angle, how might this affect the equation? What other factors (e.g., air resistance, spin, backboard) could influence the shot, and how could they be incorporated into the model?