A batter hits a pitched ball when the center of the ball is 1.35 m above the ground.The ball leaves the bat at an angle of 45° with the ground. With that launch, the ball should have a horizontal range (returning to the launch level) of 106 m. (a) Does the ball clear a 8.65-m-high fence that is 96.0 m horizontally from the launch point? (b) At the fence, what is the distance between the fence top and the ball center?
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To find out if the ball clears the fence and to determine the distance between the fence top and the ball's center, we need to analyze the motion of the ball.
First, let's break down the problem into two components: vertical and horizontal.
Vertical Motion:
1. Determine the time it takes for the ball to reach its highest point. We can use the formula:
h = ut + (1/2)gt^2, where h is the vertical displacement (1.35 m), u is the initial vertical velocity, t is the time, and g is the acceleration due to gravity (-9.8 m/s^2).
Since the ball reaches its highest point when it has no vertical velocity (at the top of its trajectory), the equation becomes:
0 = u(vert)t + (1/2)g(t^2). This equation allows us to solve for the time it takes for the ball to reach its peak.
2. Determine the time it takes for the ball to hit the ground. We can use the equation:
0 = u(vert)t + (1/2)g(t^2), where the initial vertical velocity u(vert) is equal to the initial launch velocity multiplied by the sine of the launch angle.
We can solve for t using this equation.
Horizontal Motion:
3. Calculate the horizontal distance traveled by the ball. We can use the formula:
S(horiz) = u(horiz)t, where S(horiz) is the horizontal distance (106 m) and u(horiz) is the initial horizontal velocity.
The initial horizontal velocity u(horiz) is equal to the initial launch velocity multiplied by the cosine of the launch angle.
Now, let's solve parts (a) and (b) of the problem using these steps:
(a) Does the ball clear the fence?
To determine if the ball clears the fence, we need to check if the maximum height of the ball is greater than the height of the fence. If it is, then the ball clears the fence; otherwise, it does not.
1. Calculate the time it takes for the ball to reach its highest point using the equation from step 1.
2. Calculate the maximum height reached by the ball using the equation:
h_max = -u(vert)^2 / (2g)
3. Compare the maximum height to the height of the fence. If the maximum height is greater than 8.65 m, the ball clears the fence.
(b) What is the distance between the fence top and the ball center?
1. Calculate the time it takes for the ball to travel horizontally to the fence using the equation from step 3.
2. Determine the vertical displacement of the ball at that time using the equation:
h = ut + (1/2)gt^2, where u is the initial vertical velocity (which is equal to u(vert) from earlier), t is the time, and g is the acceleration due to gravity.
3. Calculate the distance between the fence top and the ball center by subtracting the vertical displacement from the height of the fence.
By following these steps, you can find the answers to both parts (a) and (b) of the problem.
assuming there is no air friction,
A) yes
B) (10+1.35)-8.65= 2.7m