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Let's analyze each part of the question step-by-step, beginning with what has already been answered:

### Given Information
The parametric equations for the baseball's path are:
- \( x(t) = 144t \)
- \( y(t) = -16t^2 + 6t + 5 \)

These equations indicate the position in feet at time \( t \) in seconds. 

### Completed Parts:
The following answers are already provided:

1. **(a) Horizontal velocity \( x'(t) = 144 \) ft/sec.**
2. **(b) Horizontal velocity at home plate = 144 ft/sec.**
3. **(c) Vertical velocity \( y'(t) = -32t + 6 \) ft/sec.**

### Remaining Parts:

Let's go through each remaining question.

#### (d) Speed of the baseball (in mph) when it passes over home plate
The speed \( s(t) \) at any time \( t \) is given by:
\[
s(t) = \sqrt{[x'(t)]^2 + [y'(t)]^2}
\]

1. Since \( x'(t) = 144 \), this part remains constant.
2. To find the time \( t \) when it reaches home plate, we use the horizontal distance:
   \[
   x(t) = 144t = 60.5 \Rightarrow t = \frac{60.5}{144}
   \]
   Calculating \( t \) and using this to find \( y'(t) \), then plug values into the speed formula.

#### (e) Time when the baseball hits the ground
To find when the baseball hits the ground, set \( y(t) = 0 \) and solve for \( t \):
\[
-16t^2 + 6t + 5 = 0
\]

#### (f) Magnitude of the angle when the baseball hits the ground
To find the angle \( \theta \) with the ground, use:
\[
\theta = \arctan\left(\frac{|y'(t)|}{x'(t)}\right)
\]
where \( t \) is the time found in (e).

Would you like a detailed step-by-step calculation for each part, or do you have specific questions about any part? 

Here are 5 questions that could extend this discussion:
1. What role does the initial vertical velocity play in determining the ball's trajectory?
2. How would the angle of impact change if the initial vertical velocity were higher?
3. What would happen if we increased the horizontal velocity? Would the time to reach the ground change?
4. Can you explain the relationship between the vertical and horizontal components in determining the ball's final speed?
5. What real-world factors could cause variations from this theoretical path?

**Tip:** Always keep track of units, especially when converting speeds from ft/sec to mph!

A major league baseball pitcher throws a pitch following these parametric equations: x(t) = 144t and y(t) = -16t^2 + 6t + 5. The task is to calculate the horizontal and vertical velocities, the speed when it reaches home plate, the time it hits the ground, and the angle at which it hits the ground.

Explore the physics of a baseball pitch with parametric equations. Calculate the baseball's horizontal and vertical velocities, speed as it crosses home plate, time to hit the ground, and the impact angle. Suitable for college-level calculus and physics students.

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