Final answer:
To determine the angle at which a player must throw a basketball from the free throw line to hit the basket, apply principles of projectile motion. Use kinematic equations to link the horizontal and vertical components of the ball's motion to the initial speed, release height, and horizontal distance to the basket.
Step-by-step explanation:
To solve for the angle above the horizontal at which the basketball should be thrown to hit the basket from the free throw line, we need to apply the principles of projectile motion. The information given states that the free throw line is 4.57 meters from the basket, and the basket is 3.05 meters above the floor. The player releases the ball at a height of 2.44 meters with an initial speed of 8.15 meters per second.
To find the required angle, we can use the kinematic equations for projectile motion. First, we calculate the time it takes for the ball to reach the horizontal distance of 4.57 meters. The horizontal velocity component is vx = v0 cos(θ), and the horizontal distance traveled is given by the equation x = vx t. From this, we can express the time t as t = x / (v0 cos(θ)).
Next, to ensure that the ball reaches the height of the basket, we can use the vertical position equation, y = y0 + v0y t - (1/2)g t2, where y0 is the initial height of the ball, v0y is the initial vertical velocity component, t is the time which the ball is in the air, and g is the acceleration due to gravity. Setting y0 to 2.44 meters, y to 3.05 meters and using the expression for t we derived from the horizontal motion, we substitute and solve for the angle θ.
This projectile motion problem can be solved using algebra and trigonometry to find the angle that satisfies both the horizontal and vertical displacement requirements. Since the equation would be non-linear, a numerical method or graphical solution could be used to find the precise angle required.