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 13.8° below the horizontal. It is released 0.65 m above the floor. What horizontal distance does the ball cover before bouncing?
4.3
To find the horizontal distance covered by the ball before bouncing, we need to analyze the motion of the ball and determine the time it takes to reach the floor.
We can split the ball's motion into horizontal and vertical components. The initial velocity can be divided into these components using trigonometry.
The horizontal component of the initial velocity (Vx) can be determined using the initial speed (3.9 m/s) and the angle (13.8°) as follows:
Vx = initial speed * cos(angle) = 3.9 m/s * cos(13.8°) = 3.78 m/s
The vertical component of the initial velocity (Vy) can be determined using the initial speed (3.9 m/s) and the angle (13.8°) as follows:
Vy = initial speed * sin(angle) = 3.9 m/s * sin(13.8°) = 0.92 m/s
Now, we can use the vertical motion equation to find the time it takes for the ball to reach the floor:
vertical displacement = initial vertical velocity * time + (0.5 * acceleration * time^2)
Since the ball is released 0.65 m above the floor, the vertical displacement is -0.65 m (negative because the ball is moving downwards).
-0.65 m = 0.92 m/s * time + (0.5 * 9.8 m/s^2 * time^2)
Rearranging the equation:
0.5 * 9.8 m/s^2 * time^2 + 0.92 m/s * time - 0.65 m = 0
This is a quadratic equation. We can solve it using the quadratic formula:
time = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this equation, a = 0.5 * 9.8 m/s^2, b = 0.92 m/s, and c = -0.65 m.
Solving the equation, we get two values for time: time1 ≈ 0.15 s and time2 ≈ 1.02 s. Since we are interested in the time it takes for the ball to hit the ground, we'll use the larger value, time2 = 1.02 s.
Finally, to find the horizontal distance covered by the ball before bouncing, we can use the horizontal component of the velocity (3.78 m/s) and the time (1.02 s):
horizontal distance = horizontal velocity * time = 3.78 m/s * 1.02 s ≈ 3.86 m
Therefore, the ball covers approximately 3.86 meters horizontally before bouncing.