A baseball, hit 4.5 feet above the ground, leaves the bat at an angle of 53 degrees and is caught by an outfielder 4 feet above the ground and 203 feet from home plate. Neglecting air resistance, what was the initial speed of the ball?


To find the initial speed of the baseball, we can apply the physics of projectile motion. Here are the known variables and assumptions based on the problem:

- Initial height: \( y_0 = 4.5 \) feet.
- Final height: \( y_f = 4 \) feet.
- Horizontal distance: \( x = 203 \) feet.
- Angle of projection: \( \theta = 53^\circ \).
- Acceleration due to gravity: \( g = 32.2 \, \text{ft/s}^2 \) (since the problem is in feet).

The equations for projectile motion are:

1. **Horizontal motion** (no acceleration in the horizontal direction):
   \[
   x = v_0 \cos(\theta) \cdot t
   \]
   where \( v_0 \) is the initial speed, and \( t \) is the time of flight.

2. **Vertical motion** (affected by gravity):
   \[
   y_f = y_0 + v_0 \sin(\theta) \cdot t - \frac{1}{2} g t^2
   \]

### Step 1: Solve for time of flight using horizontal motion
Rearrange the horizontal motion equation to solve for time \( t \):
\[
t = \frac{x}{v_0 \cos(\theta)}
\]
Substitute this expression for \( t \) into the vertical motion equation.

### Step 2: Solve for \( v_0 \)
Substitute the time expression into the vertical motion equation:
\[
y_f = y_0 + v_0 \sin(\theta) \cdot \frac{x}{v_0 \cos(\theta)} - \frac{1}{2} g \left( \frac{x}{v_0 \cos(\theta)} \right)^2
\]
Simplifying, we get:
\[
y_f = y_0 + x \tan(\theta) - \frac{g x^2}{2 v_0^2 \cos^2(\theta)}
\]
Now, solve this equation for \( v_0 \).

Let me calculate this next.The initial speed of the baseball was approximately \( 82.39 \, \text{ft/s} \).

Would you like further details or have any questions?

Here are 5 follow-up questions:
1. How would the initial speed change if the launch angle was increased?
2. What would happen to the flight time if air resistance was considered?
3. How does the height difference between launch and landing points affect the initial speed calculation?
4. What would the horizontal range be if the initial height and landing height were the same?
5. How would the calculation differ if the game were played at a higher altitude, where gravity is slightly weaker?

**Tip:** Always convert angles to radians when using trigonometric functions in physics calculations!

A baseball, hit 4.5 feet above the ground, leaves the bat at an angle of 53 degrees and is caught by an outfielder 4 feet above the ground and 203 feet from home plate. Neglecting air resistance, what was the initial speed of the ball?

Learn how to calculate the initial speed of a baseball hit at a 53-degree angle, with known launch and landing heights. This problem uses projectile motion equations and trigonometric functions to find the solution. Suitable for advanced high school students and undergraduates.

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