A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115ft/s at an angle 50 above the horizontal. Is it a home run? (In other words, does the ball clear the fence?)

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Section 14.4, Problem 28 - Illinois

You will see solution with explanation.

Recall your equation of motion

y(x) = 3 + (tanθ)x - g/(2(v cosθ)^2) x^2
Now just plug in your numbers and find whether
y(400) > 10

To determine if the ball clears the fence, we can analyze its vertical motion.

First, let's break down the initial velocity of the ball into its horizontal and vertical components using trigonometry. The vertical component can be found by multiplying the initial speed (115 ft/s) by the sine of the launch angle (50°), and the horizontal component can be found by multiplying the initial speed by the cosine of the launch angle.

Vertical component (V_y) = 115 ft/s * sin(50°)
Horizontal component (V_x) = 115 ft/s * cos(50°)

Next, we need to analyze the time it takes for the ball to reach its highest point. The time to reach the maximum height can be found using the formula:

Time (t) = (final velocity) / (acceleration)
Since the final velocity at the maximum height is 0 ft/s (the ball momentarily stops moving vertically), we can substitute that into the formula:
0 = V_y - g*t

Using the value of the acceleration due to gravity (g) = 32.2 ft/s², we can solve for t.

0 = V_y - (32.2 ft/s²)*t
t = V_y / (32.2 ft/s²)

Now, we can find the maximum height (H_max) the ball reaches by substituting the value of t:

H_max = V_y*t - (1/2)*g*t²

Note: The above equation assumes that the vertical displacement is 0 at the highest point. However, in this case, the vertical displacement should be 3 ft above the ground.

So, the actual maximum height reached by the ball (H_actual) would be H_max + 3 ft.

Once we have the maximum height, we can analyze whether it clears the 10 ft high fence by comparing it to the fence height. If the H_actual is greater than 10 ft, it will clear the fence and be a home run.

Now, let's calculate the maximum height and check if it's a home run.

Given:
Initial speed (V) = 115 ft/s
Launch angle (θ) = 50°
Fence height (H_fence) = 10 ft
Horizontal distance (D) = 400 ft

Using the equations mentioned above, we can substitute the given values to calculate the maximum height:

V_y = 115 ft/s * sin(50°)
V_x = 115 ft/s * cos(50°)

t = V_y / (32.2 ft/s²)

H_max = V_y * t - (1/2) * g * t²
H_actual = H_max + 3 ft

Now, compare the H_actual with H_fence:

If H_actual > H_fence, then it's a home run.
If H_actual ≤ H_fence, then it doesn't clear the fence and is not a home run.

By substituting the values into the equations and calculating, you can determine if it's a home run or not.