A basketball player who is 2.00 m tall is standing on the floor 8 m from the basket. If he shoots the ball at a 390 angle with the horizontal, what initial speed must he shoot the ball so that it goes through the hoop without striking the blackboard. The basket height is 3.05 m.
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To solve this problem, we can break it down into two main components: horizontal motion and vertical motion. We'll analyze each component separately and then combine the results to find the answer.
1. Horizontal Motion:
The horizontal motion of the ball is not affected by gravity, so it moves at a constant velocity. Let's find the time it takes for the ball to reach the basket using the horizontal distance.
Distance = Speed x Time
The distance from the player to the basket (horizontal distance) is given as 8 m.
Time = Distance / Speed
Since the horizontal distance is constant, we can assume the player shoots the ball with the same horizontal speed throughout its trajectory.
2. Vertical Motion:
The vertical motion of the ball is affected by gravity. We need to consider the initial height of the player, the height of the basket, and the angle at which the ball is shot.
It's given that the player is 2.00 m tall, and the height of the basket is 3.05 m. Therefore, the total vertical distance the ball needs to cover is:
Vertical Distance = Height of basket - Height of player = 3.05 m - 2.00 m = 1.05 m
The vertical motion of the ball can be describe using the equation of motion:
Vertical Distance = (Initial Vertical Velocity x Time) + (0.5 x Acceleration x Time^2)
As we know the angle at which the ball is shot, we can break down the initial velocity into its horizontal and vertical components.
Initial Vertical Velocity = Initial Speed x sin(angle)
Initial Horizontal Velocity = Initial Speed x cos(angle)
The acceleration in this case is the acceleration due to gravity, which is approximately 9.8 m/s^2.
3. Combining the Results:
We can now use the time calculated from the horizontal motion to determine the required initial vertical velocity.
Substituting the values into the vertical motion equation:
1.05 m = (Initial Speed x sin(39°) x Time) - (0.5 x 9.8 m/s^2 x Time^2)
Using the time calculated from the horizontal motion, substitute it into the above equation:
1.05 m = (Initial Speed x sin(39°) x (8 m / Initial Speed x cos(39°))) - (0.5 x 9.8 m/s^2 x (8 m / Initial Speed x cos(39°))^2)
Simplifying this equation will give us the initial speed required to shoot the ball.
Finally, we solve this equation using mathematical techniques such as substitution or factoring to find the initial speed of the ball.