A 2.12 m tall basketball player wants to sink the ball into a basket located 14 m from the player, as in Figure P3.58. If he shoots the ball at a 46° angle, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard?
Vo^2*sin(2A)/g = 14 m.
Vo^2*sin(92)/9.8 = 14
0.10198*Vo^2 = 14
Vo^2 = 137.3
Vo = 11.72 m/s.
v
To find the initial speed, we can use the equations of projectile motion.
Let's start by analyzing the vertical motion of the basketball. The ball's motion in the vertical direction can be described by the equation:
h = (v₀y * t) - (0.5 * g * t²)
where h is the vertical distance (height) the ball needs to cover, v₀y is the initial vertical velocity of the ball, t is the time of flight, and g is the acceleration due to gravity (approximately 9.8 m/s²).
In this case, we want the ball to go through the hoop without striking the backboard, which means the vertical distance traveled by the ball should be the height of the hoop. Given that the basketball player is 2.12 m tall, and the basketball hoop is typically placed about 3.05 m above the ground, the height the ball needs to cover is:
h = 3.05 m + 2.12 m = 5.17 m
Now let's analyze the horizontal motion of the basketball. The ball's motion in the horizontal direction is described by the equation:
d = v₀x * t
where d is the horizontal distance traveled by the ball, v₀x is the initial horizontal velocity of the ball, and t is the time of flight.
In this case, the horizontal distance is given as 14 m.
We also know that the angle of projection is 46°. From this, we can determine the initial vertical and horizontal velocities:
v₀x = v₀ * cos(46°)
v₀y = v₀ * sin(46°)
where v₀ is the initial speed of the ball.
Now, we need to find the time of flight, which is the same for both the vertical and horizontal motions. We can use the vertical motion equation to find the time:
h = (v₀y * t) - (0.5 * g * t²)
Rearranging the equation, we get:
0.5 * g * t² - (v₀y * t) + h = 0
This is a quadratic equation in terms of t. We can solve it using the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
In this case, a = 0.5 * g, b = -v₀y, and c = h.
Now, we can substitute the values and solve for t. Once we have the time of flight, we can use it to find the initial speed using the horizontal motion equation:
v₀ = d / (v₀x * t)
Substituting the values, we can calculate the initial speed required for the basketball player to sink the ball into the hoop without striking the backboard.