The free throw line in basketball is 4.57 m (15 ft) from the basket, which is 3.05 m (10 ft) above the floor. A player standing on the free throw line throws the ball with an initial speed of 8.00 m/s, releasing it at a height of 2.48 m above the floor. At what angle above the horizontal must the ball be thrown to exactly hit the basket? Note that most players will use a large initial angle rather than a flat shot because it allows for a larger margin of error.

° above the horizontal

Ignoring ball and hoop diameter, just solve

y(x) = -gsec^2(θ)/2v^2 x^2 + tanθ x + h
when y(4.57) = 3.05

-9.8 sec^2θ/(2*8^2) * 4.57^2 + tanθ (4.57) + 2.48 = 3.05

-1.6 (1+tan^2θ) + 4.57 tanθ - 0.57 = 0
1.6 tan^2θ - 4.57 tanθ + 2.17 = 0

tanθ = .602 or 2.255
θ = 31° or 66°

Steve,

Do you mind explaining where you derived the y(x) equation?
It looks like something I had before I looked for help on this problem and I am just trying to clarify just in case I see this problem again.
I did use this help to get the problem answered correctly and am just curious now.
Any feedback would be great

Well, I have to say, this basketball player is really putting some thought into their shot! They must be shooting for the stars, or at least the basket. Now, let's calculate that angle.

First, let's figure out the range, which is the horizontal distance from the player to the basket. We'll use the kinematic equation:

Range = (Initial speed)^2 * sin(2θ) / g

where g is the acceleration due to gravity. Since we want the ball to hit the basket, the range should be equal to the distance from the free throw line to the basket, which is 4.57 m.

Now, let's solve for θ. But wait, before we do that, why don't we take a moment to appreciate the beauty of this equation? It's really something special.

Alright, now let's solve for θ. We rearrange the equation to get:

sin(2θ) = (Range * g) / (Initial speed)^2

Taking the inverse sin of both sides gives us:

2θ = arcsin[(Range * g) / (Initial speed)^2]

And finally, divide by 2 to get θ:

θ = (1/2) * arcsin[(Range * g) / (Initial speed)^2]

So, if we plug in the values:
Range = 4.57 m
g = 9.8 m/s^2
Initial speed = 8.00 m/s

We get:

θ ≈ (1/2) * arcsin[(4.57 * 9.8) / (8.00)^2]

Now, let me calculate that for you. Drumroll, please...

Calculating... calculating...

Okay, according to my calculations, the angle above the horizontal at which the ball must be thrown to hit the basket is approximately 47.3°.

Now, remember, this is just a calculation, and in real life, there are many other factors at play, like air resistance, the player's skill, and the ever-present possibility of a rogue squirrel intercepting the ball. So, take it with a grain of salt and give it your best shot!

To find the angle above the horizontal at which the ball must be thrown to hit the basket, we can use the principles of projectile motion. Here's how you can approach it:

Step 1: Identify the known values:
- Initial speed (v₀) = 8.00 m/s
- Initial height (y₀) = 2.48 m
- Horizontal distance (x) = 4.57 m
- Vertical distance (y) = 3.05 m

Step 2: Write down the relevant equations:
We can use the equations of projectile motion to determine the angle. The two equations we require are:
- Vertical displacement equation: y = y₀ + v₀sinθt - 0.5gt²
- Horizontal displacement equation: x = v₀cosθt

Step 3: Solve the horizontal displacement equation for time (t):
Rearrange the equation to solve for t:
t = x / (v₀cosθ)

Step 4: Substitute the expression for time (t) into the vertical displacement equation:
Plug in the value of t from step 3 into the vertical displacement equation:
y = y₀ + v₀sinθ (x / (v₀cosθ)) - 0.5g [(x / (v₀cosθ))]²

Step 5: Simplify the equation:
Combine like terms in the equation and simplify if possible.

Step 6: Rearrange the equation to solve for θ:
Rearrange the equation to solve for θ. Here, you may need to use math software or calculators to solve the equation. The equation may be in a trigonometric form, such as sinθ, so you may need to use inverse trigonometric functions (e.g., arcsine) to find the angle.

By following these steps, you can determine the angle above the horizontal at which the ball must be thrown to hit the basketball hoop.