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.

Question

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.

Answer 1

See previous post.

Answer 2

To determine if the ball clears the fence, we need to find the vertical distance the ball reaches when it reaches the wall.

To solve this problem, we can break down the initial velocity into its horizontal and vertical components.

Given:
Initial velocity (V) = 103 feet per second
Angle of elevation (θ) = 34 degrees
Horizontal wind velocity (Vw) = 22 feet per second
Height of the wall (H) = 10 feet
Distance from home plate to the wall (D) = 300 feet

First, let's find the horizontal and vertical components of the initial velocity:

Horizontal component (Vx) = V * cos(θ)
Vertical component (Vy) = V * sin(θ)

Vx = 103 * cos(34°) = 85.43 feet per second (approx.)
Vy = 103 * sin(34°) = 55.50 feet per second (approx.)

Now, we need to find the time it takes for the ball to reach the wall. Since there is horizontal wind blowing, the time taken can be found by dividing the horizontal distance (D) by the sum of the horizontal component of the initial velocity (Vx) and the horizontal wind velocity (Vw):

Time (t) = D / (Vx + Vw)
t = 300 / (85.43 + 22) = 3.161 seconds (approx.)

Next, we can find the maximum height reached by the ball. To do this, we'll use the kinematic equation for vertical motion:

H = Vy * t + 0.5 * g * t²

Since the ball is hit vertically, the initial vertical velocity (Vy) remains constant throughout its flight. We can neglect air resistance and assume the acceleration due to gravity (g) is approximately 32.2 feet per second squared.

H = 55.50 * 3.161 + 0.5 * 32.2 * 3.161²
H ≈ 175.32 feet

The maximum height reached by the ball is approximately 175.32 feet.

Finally, we can determine if the ball clears the wall. If the height reached by the ball (H) is greater than the height of the wall (10 feet), then the ball clears the fence.

Since 175.32 feet is greater than 10 feet, we can conclude that the ball clears the fence.