What vertical velocity component should the pitcher use so that the pitch crosses home plate exactly 4 ft above the​ ground?

To determine the vertical velocity component that the pitcher should use, we need to break down the problem with the following assumptions and known variables:

1. **Displacement in the vertical direction** \(\Delta y\): The ball must cross the home plate at a height of 4 feet above the ground.
2. **Initial height of the ball**: Typically, a pitch starts at about 6 feet off the ground (this can vary, but we assume it's 6 feet for this example).
3. **Acceleration due to gravity** \(-g\): Gravity acts downward with \(g = 32.2 \, \text{ft/s}^2\).
4. **Time of flight** \(t\): This is the time it takes for the ball to travel horizontally from the pitcher to the home plate, which depends on the horizontal velocity and distance between the pitcher and home plate (typically 60.5 feet).

We'll solve for the vertical velocity \(v_{y_0}\) that the pitcher needs so that the ball reaches 4 feet when it crosses the plate.

### Step-by-Step Process

1. **Equation of vertical motion**:
   The vertical position of the ball at any time \(t\) is given by the kinematic equation:
   \[
   y = y_0 + v_{y_0}t - \frac{1}{2} g t^2
   \]
   where:
   - \(y\) is the final height (4 ft),
   - \(y_0\) is the initial height (6 ft),
   - \(v_{y_0}\) is the vertical component of the initial velocity,
   - \(t\) is the time of flight,
   - \(g\) is the acceleration due to gravity (32.2 ft/s\(^2\)).

2. **Solve for \(v_{y_0}\)**:
   We can rearrange this equation to solve for the vertical velocity component, but we first need to know the time of flight, which depends on the horizontal velocity and distance. 

Would you like me to proceed with the calculation by assuming a horizontal velocity, or would you like to specify the time or horizontal velocity of the pitch? Feel free to upload any further information if needed!

What vertical velocity component should the pitcher use so that the pitch crosses home plate exactly 4 ft above the ground?

Learn how to calculate the vertical velocity component required for a baseball pitch to cross home plate 4 feet above the ground using kinematic equations. This step-by-step guide covers projectile motion and gravity, suitable for high school physics students in Grades 11-12.

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