A baseball is thrown directly upward (in z direction) from the ground (z = 0) with some initial velocity. Assume that air drag is negligible (e.g. ball travels in a vacuum).
A baseball is thrown directly upward (in z direction) from the ground (z = 0) with some initial velocity. Assume that air drag is negligible (e.g. ball travels in a vacuum).
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![### Physics Problem: Maximum Height of a Baseball
**Problem Description:**
A baseball is thrown directly upward (in the z direction) from the ground (z = 0) with some initial velocity. Assume that air drag is negligible (e.g., the ball travels in a vacuum).
**Task:**
Fill in the following table.
**Dependent Variable:**
The dependent variable in this problem is the maximum height of the baseball, \( z_{\text{max}} \).
**Table:**
| Independent Variable | If the Independent Variable (with all other parameters held constant) | Your Prediction From Physical Intuition:<br>Does \( z_{\text{max}} \) increase (↑) or decrease (↓) ? | Analysis Prediction: | Does Your Prediction Match the Analysis Prediction: |
|-------------------------------------|-----------------------------------------------------------------------|------------------------------------------------------------------------------------------------------|----------------------|------------------------------------------------------|
| Initial Ball Velocity \( V_0 \) | ↑ | | | |
| Gravitational acceleration \( g \) | ↑ | | | |
**Questions to be addressed:**
b) Find an expression for the instantaneous baseball velocity \( V(t) \).
c) Find an expression for the instantaneous baseball height \( z(t) \).
d) Find an expression for the maximum height of the baseball, \( z_{\text{max}} \).
### Detailed Instructions:
#### Part (b): Velocity as a Function of Time
To find the expression for the instantaneous velocity \( V(t) \) of the baseball:
- Use Newton’s second law of motion under constant acceleration (gravity).
- The formula for the velocity at any time \( t \) after the initial throw is:
\[ V(t) = V_0 - gt \]
where:
- \( V_0 \) is the initial velocity.
- \( g \) is the acceleration due to gravity (assumed to be 9.8 m/s² downward).
#### Part (c): Height as a Function of Time
To find the expression for the height \( z(t) \) of the baseball:
- Use the kinematic equations for motion under constant acceleration.
- The height at any time \( t \) is given by:
\[ z(t) = V_0 t - \frac{1}{2} g t^2 \]
#### Part (d): Maximum Height
To find the maximum height \( z_{\text{max}} \):
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F508ae4f5-8e9f-43cb-9ab2-37d719369fa8%2Fa0d33192-4a24-4ad0-862e-eb39764c594b%2Fugi83yg.png&w=3840&q=75)
Transcribed Image Text:### Physics Problem: Maximum Height of a Baseball
**Problem Description:**
A baseball is thrown directly upward (in the z direction) from the ground (z = 0) with some initial velocity. Assume that air drag is negligible (e.g., the ball travels in a vacuum).
**Task:**
Fill in the following table.
**Dependent Variable:**
The dependent variable in this problem is the maximum height of the baseball, \( z_{\text{max}} \).
**Table:**
| Independent Variable | If the Independent Variable (with all other parameters held constant) | Your Prediction From Physical Intuition:<br>Does \( z_{\text{max}} \) increase (↑) or decrease (↓) ? | Analysis Prediction: | Does Your Prediction Match the Analysis Prediction: |
|-------------------------------------|-----------------------------------------------------------------------|------------------------------------------------------------------------------------------------------|----------------------|------------------------------------------------------|
| Initial Ball Velocity \( V_0 \) | ↑ | | | |
| Gravitational acceleration \( g \) | ↑ | | | |
**Questions to be addressed:**
b) Find an expression for the instantaneous baseball velocity \( V(t) \).
c) Find an expression for the instantaneous baseball height \( z(t) \).
d) Find an expression for the maximum height of the baseball, \( z_{\text{max}} \).
### Detailed Instructions:
#### Part (b): Velocity as a Function of Time
To find the expression for the instantaneous velocity \( V(t) \) of the baseball:
- Use Newton’s second law of motion under constant acceleration (gravity).
- The formula for the velocity at any time \( t \) after the initial throw is:
\[ V(t) = V_0 - gt \]
where:
- \( V_0 \) is the initial velocity.
- \( g \) is the acceleration due to gravity (assumed to be 9.8 m/s² downward).
#### Part (c): Height as a Function of Time
To find the expression for the height \( z(t) \) of the baseball:
- Use the kinematic equations for motion under constant acceleration.
- The height at any time \( t \) is given by:
\[ z(t) = V_0 t - \frac{1}{2} g t^2 \]
#### Part (d): Maximum Height
To find the maximum height \( z_{\text{max}} \):
-
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