Use the model for projectile motion, assuming there is no air resistance and g = 32 feet per second per second.
A baseball player at second base throws a ball 90 feet to the player at first base. The ball is released at a point 5 feet above the ground with an initial speed of 50 miles per hour and at an angle of 19° above the horizontal. At what height (in ft) does the player at first base catch the ball? (Round your answer to three decimal places.)

To solve this problem, we'll use the equations of projectile motion to find the height at which the player at first base catches the ball. Here are the key steps:

### Given:
- Distance between the two players (horizontal distance): \( d = 90 \, \text{ft} \)
- Initial speed of the ball: \( v_0 = 50 \, \text{mph} \)
- Launch angle: \( \theta = 19^\circ \)
- Initial height: \( h_0 = 5 \, \text{ft} \)
- Gravitational acceleration: \( g = 32 \, \text{ft/s}^2 \)

First, we need to convert the initial speed from miles per hour (mph) to feet per second (ft/s):

\[
v_0 = 50 \, \text{mph} \times \frac{5280 \, \text{ft}}{1 \, \text{mile}} \times \frac{1 \, \text{hr}}{3600 \, \text{sec}} = 73.333 \, \text{ft/s}
\]

### Step 1: Horizontal motion equation
The horizontal motion is governed by:

\[
x(t) = v_0 \cos(\theta) t
\]

We can solve for the time, \( t \), when the ball reaches first base (i.e., \( x = 90 \, \text{ft} \)):

\[
90 = 73.333 \cos(19^\circ) t
\]

### Step 2: Vertical motion equation
The vertical motion is governed by:

\[
y(t) = h_0 + v_0 \sin(\theta) t - \frac{1}{2} g t^2
\]

Now, we'll solve for the time of flight and substitute it into the vertical equation to find the height at which the ball is caught. I'll calculate it step-by-step.The player at first base catches the ball at a height of approximately \( 9.033 \, \text{ft} \) above the ground.

Would you like any further details or explanations?

Here are five related questions you could explore:
1. How would the answer change if the initial speed were different?
2. What if the angle of projection was steeper, say 30°? How would that affect the outcome?
3. How long does it take for the ball to reach first base?
4. What is the maximum height the ball reaches during its flight?
5. How does air resistance impact projectile motion in real-world scenarios?

**Tip:** In projectile motion problems, always break the motion into horizontal and vertical components to simplify calculations.

A baseball player at second base throws a ball 90 feet to the player at first base. The ball is released at a point 5 feet above the ground with an initial speed of 50 miles per hour and at an angle of 19° above the horizontal. At what height (in ft) does the player at first base catch the ball? (Round your answer to three decimal places.)

Learn how to calculate the height at which a baseball is caught using projectile motion equations. This problem involves a 90-foot throw with an initial speed of 50 mph and an angle of 19°. Key topics include kinematics and trigonometric decomposition. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation