In a game of basketball, a forward makes a bounce pass to the center. The ball is thrown with an initial speed of 3.9 m/s at an angle of 25 degrees above the horizontal. It is released 0.75 m above the floor.
What horizontal distance does the ball cover before bouncing?
Break up the initial velocity in to horizontal and vertical components.
vertical
Vi=3.9sin25
hf=ho+vi*t-4.9t^2
0=.75+3.9sin25*t-4.9t^2 solve for time in air, t.
Then
Horizontal
distance=3.9cos25*t
To calculate the horizontal distance covered by the ball before bouncing, we need to analyze the projectile motion of the ball.
Step 1: Separate the motion into horizontal and vertical components. The initial velocity can be broken down into its horizontal (Vx) and vertical (Vy) components.
Given:
Initial speed (Vi) = 3.9 m/s
Launch angle (θ) = 25 degrees above the horizontal
Vy = Vi * sin(θ)
Vx = Vi * cos(θ)
Step 2: Calculate the time taken for the ball to reach the floor. Since the ball is released 0.75 m above the floor, the vertical displacement is -0.75 m (negative because it is going downwards). We can use the kinematic equation:
Δy = Vy * t - (1/2) * g * t^2
where:
Δy = vertical displacement
g = acceleration due to gravity = 9.8 m/s^2
t = time taken
Using -0.75 m for Δy:
-0.75 m = (Vi * sin(θ)) * t - (1/2) * g * t^2
This equation is a quadratic equation in terms of t. We can solve it to find the time taken.
Step 3: Calculate the horizontal distance covered by the ball before bouncing.
The horizontal distance (dx) can be calculated using:
dx = Vx * t
Now, let's calculate each component step by step:
1. Calculate the vertical component:
Vy = 3.9 m/s * sin(25 degrees) = 1.644 m/s
2. Calculate the time taken:
-0.75 m = (1.644 m/s) * t - (1/2) * (9.8 m/s^2) * t^2
This quadratic equation can be solved using various techniques like factoring, quadratic formula, or graphing. Solve it to find the time taken (t).
3. Calculate the horizontal distance:
Vx = 3.9 m/s * cos(25 degrees) = 3.52 m/s
dx = 3.52 m/s * t
Substitute the value of t to find the horizontal distance covered by the ball before bouncing.