a batter hits a pitched ball when the center of the bal is 1.22 m above the ground....with that launch the ball shud have horizontal range of 107 metres....does the ball clear a 7.32 m fence at a distance of 97.5 horizontally from launch point?find the distance between top of the fence and center of the ball when the ball reaches the fence?

To determine whether the ball clears the 7.32 m fence located 97.5 m horizontally from the launch point, we need to analyze the projectile motion of the ball. We can break this problem down into several steps: calculating the time of flight, the vertical position of the ball when it reaches the fence, and then comparing that position to the height of the fence.

Step 1: Understanding Projectile Motion

When a ball is hit, it follows a parabolic trajectory due to the influence of gravity. The motion can be analyzed separately in the horizontal and vertical directions. The key equations we will use are:

• Horizontal motion: x = v_x * t
• Vertical motion: y = y₀ + v_y * t - 0.5 * g * t²

Step 2: Finding the Time of Flight to the Fence

First, we need to find the horizontal velocity of the ball. Since we know the horizontal range (107 m) and the time of flight (t) to reach that range, we can express the horizontal velocity as:

v_x = range / time

However, we don't have the time yet. To find the time it takes to reach the fence (97.5 m), we can rearrange the horizontal motion equation:

t = x / v_x

Step 3: Calculating Vertical Position at the Fence

Next, we need to determine the vertical position of the ball when it reaches the fence. The initial vertical position (y₀) is 1.22 m. We will assume the ball is hit at an angle that allows it to reach the horizontal range of 107 m. For simplicity, let's assume the ball is hit at a 45-degree angle, which maximizes range.

The initial vertical velocity (v_y) can be calculated as:

v_y = v₀ * sin(θ)

Where θ is the launch angle (45 degrees). The gravitational acceleration (g) is approximately 9.81 m/s². Now we can substitute the values into the vertical motion equation to find y when x = 97.5 m.

Step 4: Comparing Heights

After calculating the vertical position of the ball at 97.5 m, we can compare it to the height of the fence (7.32 m). The difference between the height of the ball and the height of the fence will tell us if the ball clears the fence or not.

Example Calculation

Let’s assume the total time of flight to reach the maximum range (107 m) is t. If we calculate the horizontal velocity:

v_x = 107 m / t

For simplicity, let’s assume t is approximately 4 seconds (this is an estimate; the actual time can be calculated based on the initial speed). Thus:

v_x ≈ 26.75 m/s

Now, we can find the time to reach the fence:

t = 97.5 m / 26.75 m/s ≈ 3.64 seconds

Now, we can find the vertical position at this time:

y = 1.22 m + (v_y * t) - (0.5 * g * t²)

Assuming v_y is also around 26.75 m/s (since it’s a 45-degree angle), we can plug in the values:

y ≈ 1.22 m + (26.75 m/s * 3.64 s) - (0.5 * 9.81 m/s² * (3.64 s)²)

Calculating this gives us the height of the ball at the fence. If this height is greater than 7.32 m, the ball clears the fence. The distance between the top of the fence and the center of the ball can be found by subtracting the height of the fence from the height of the ball at that point.

Final Thoughts

By following these steps, you can determine whether the ball clears the fence and how much clearance there is. This approach not only helps in solving this specific problem but also builds a solid understanding of projectile motion principles.