Find the ideal forward speed. Suppose here that our basket ball player is 6' 6" tall, and the ball is thrown with an upward velocity of 18 ft/sec towards a basket 13.75 feet away at 10 feet high.

Question

1. Find a height function. Use the given information to write down a definition for the function f(x) that gives the height of the ball as a function of its horizontal distance x from the shooter. Then, use the equation to find the x-coordinate of the vertex of the graph of f(x). (The value of s will remain unknown here.)
2. Use function notation to set up an equation that represents the "perfect shot"—that is, when the ball arrives at the center of the basket. (Some things to consider here: What is the function's input? What is its output? What do they represent physically?)
3. Find the ideal forward speed. Given the function f(x) you found, what forward speed s is required for a perfect shot? (Give at least 3 digits after the decimal point in your final answer.)

Answer 1

The equation of motion for the vertical position, considering gravity (−16 ft/sec²), is:

y(t) = −16t² + v₀t + h₀

where:

1. v₀​ = 18 ft/sec (initial upward velocity),
2. h₀ = 6.5 ft (initial height),
3. y(t) is the vertical height of the ball after time t.

The horizontal distance x from the player is given by:

x(t) = s ⋅ t

where s is the forward speed. Solve for t:

t = x / s

Substitute this into y(t):

f(x) = -16(x / s)² + 18(x / s) + 6.5

The height function is quadratic in x:

f(x) = -(16 / s²)x² + (18 / s)x + 6.5

The x-coordinate for the vertex for any quadratic function is x = -b / (2a)

x = (9/8)s

For a perfect shot, the ball must reach a height of 10 ft when the horizontal distance x is 13.75 ft. Set f(13.75) = 10 and solve for s. Use the quadratic formula or some other method to solve for s. When you do this, s is approximately 15.714 ft/sec, which is the ideal forward speed for a perfect shot.

Hope this was helpful.