An h1 = 2.10 m tall basketball player wants to make a basket from a distance d = 19.2 m, as seen in the figure. If he shoots the ball at α = 54.4° angle, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard? The height of the basketball hoop is h2 = 3.05 m.
3.05 = -1/2 g t^2 + v sin(α) t + 2.10
19.2 = v cos(α) t
plug in sin and cos values
solve the system for v (and t)
To find the initial speed the basketball player must throw the ball, we can use principles of projectile motion. Let's break down the problem step by step:
1. Determine the initial vertical velocity component (Vy):
In projectile motion, when the ball reaches its peak height, the vertical velocity becomes zero. Therefore, we can calculate the initial vertical velocity component (Vy) using the equation:
Vy = V * sin(α)
Where V is the initial velocity (which we need to find), and α is the launch angle (given as 54.4°).
2. Determine the time taken to reach the maximum height (t_peak):
The time taken to reach the maximum height (t_peak) can be determined using the equation:
t_peak = Vy / g
Where g is the acceleration due to gravity (approximately 9.8 m/s²).
3. Determine the maximum height reached (H):
The maximum height reached by the basketball can be calculated using the equation:
H = (Vy²) / (2g)
4. Determine the horizontal distance traveled (X):
The horizontal distance traveled by the basketball can be determined using the equation:
X = V * cos(α) * t_peak
Since the only vertical motion experienced by the ball is when it reaches its peak height, the time of flight (t_flight) is doubled to find the total time taken for the ball to reach the hoop.
Therefore, t_flight = 2 * t_peak
The horizontal distance traveled (X) can now be calculated using X = V * cos(α) * t_flight.
5. Calculate the required initial speed (V):
To determine the initial speed required for the basketball to go through the hoop, without striking the backboard, we set H + h1 = h2. Rearranging the equation, we get:
V = sqrt((X² * g) / (2 * (h2 - h1))
Where X is the horizontal distance traveled, h1 is the initial height of the basketball player, and h2 is the height of the basketball hoop.
Now we can plug in the given values and calculate the initial speed (V).