A major leaugue baseball player hits a ball 3 feet above the ground with a velocity of 103 feet per second in the direction of a 10 foot wall that is 300 feet from home plate. If the hit is at an angle of elevation of 34 degrees and there is wind blowing 22 feet per second in the SAME direction horizontally,, determine if the ball clears the fence.
See previous post.
To determine if the ball clears the fence, we need to analyze its trajectory. Let's break down the problem step by step.
1. Vertical Motion: The ball is initially hit 3 feet above the ground with an initial vertical velocity. We can use the kinematic equation for vertical motion:
vf^2 = vi^2 + 2gΔy
Where:
vf = final vertical velocity (unknown)
vi = initial vertical velocity (given as 103 ft/s at an angle of 34 degrees)
g = acceleration due to gravity (approximately 32.2 ft/s^2)
Δy = displacement in the vertical direction (given as -3 ft since we want to determine if the ball clears a 10 ft wall)
Rearranging the equation, we have:
vf^2 = vi^2 + 2gΔy
vf^2 = (103 ft/s)^2 + 2 * 32.2 ft/s^2 * (-3 ft)
Solving this equation will give us the final vertical velocity of the ball.
2. Horizontal Motion: The ball also experiences horizontal motion due to the wind blowing in the same direction. We can use the horizontal velocity of the ball, which is given as 103 ft/s cos(34 degrees) - 22 ft/s (wind speed).
The horizontal displacement of the ball can be calculated using the equation:
Δx = v * t
Δx = (103 ft/s cos(34 degrees) - 22 ft/s) * t
Where:
Δx = horizontal displacement (unknown, we need to find the time it takes to reach the fence)
v = horizontal velocity of the ball (given)
t = time (unknown)
3. Combining Vertical and Horizontal Motion:
We can use the calculated final vertical velocity (vf) to determine the time it takes for the ball to reach the top of its trajectory. At the top, the vertical velocity becomes zero, so we can use the equation:
vf = vi + gt
Solve this equation to find the time taken (t1) to reach the maximum height.
With the time taken to reach the maximum height, we can double it to get the total time (t2) taken to reach the fence since the same amount of time is taken for the ball to descend from the maximum height to the fence height.
4. Substituting time values into the horizontal displacement equation, we can calculate the total horizontal displacement (Δx) when the ball reaches the fence.
5. Compare the calculated horizontal displacement (Δx) with the distance from home plate to the fence (300 ft). If Δx is greater than or equal to 300 ft, the ball clears the fence. Otherwise, it falls short.
Follow these steps to calculate if the ball clears the fence based on the given parameters.