A 1.8 tall basketball player attempts a goal
11 from the basket that is 3.05 high.If he shoots the ball at a 54
◦
angle, at what
initial speed must he throw the basketball so
that it goes through the hoop without striking
the backboard? The acceleration of gravity is
9.81 m/s
2
.
Answer in units of m/s
5dythftf
To find the initial speed at which the basketball player must throw the basketball, we can use the principles of projectile motion. Let's break down the steps:
Step 1: Identify the given information:
- Height of the basketball player: 1.8 m
- Distance from the basket: 11 m
- Height of the basket: 3.05 m
- Angle of the shot: 54 degrees
- Acceleration due to gravity: 9.81 m/s^2
Step 2: Find the horizontal and vertical components of the initial velocity:
To calculate the initial velocity, we need to break it down into horizontal and vertical components. We can use trigonometry to find these components.
The horizontal component can be determined using the formula:
Vx = V * cos(theta)
where Vx is the horizontal component and theta is the angle of the shot.
The vertical component can be determined using the formula:
Vy = V * sin(theta)
where Vy is the vertical component and theta is the angle of the shot.
Step 3: Determine the time taken to reach the basket:
Next, we need to find the time it takes for the basketball to reach the basket. We can use the equation:
y = Vy * t + (1/2) * g * t^2
where y is the vertical distance (height of the basket - height of the player), Vy is the vertical component of the initial velocity, g is the acceleration due to gravity, and t is the time taken to reach the basket.
Step 4: Calculate the initial speed:
To find the initial speed V, we can use the equation:
x = Vx * t
where x is the horizontal distance (distance from the basket), Vx is the horizontal component of the initial velocity, and t is the time taken to reach the basket.
Let's put it all together and calculate the answer:
1. Calculate the horizontal and vertical components of the initial velocity:
Vx = V * cos(theta) = V * cos(54)
Vy = V * sin(theta) = V * sin(54)
2. Determine the time taken to reach the basket:
y = Vy * t + (1/2) * g * t^2
(3.05 - 1.8) = (V * sin(54)) * t - (1/2) * (9.81) * t^2
3. Solve for t:
(3.05 - 1.8) = V * sin(54) * t - (4.905) * t^2
4. Calculate the horizontal distance (x):
x = Vx * t = V * cos(54) * t
5. Solve for V:
Use the given horizontal distance (x = 11) to substitute into the equation from the previous step:
11 = V * cos(54) * t
Substitute the value of t from step 3 into this equation and solve for V.
By following these steps, you should be able to find the initial speed at which the basketball player must throw the basketball.