Using the information given in the image, find: a) the velocity at 2 secondsb) when it will reach the maximum heightc) what the maximum height isd) the acceleration at any time (t)e) when it will hit the groundf) the velocity at which it will hit the ground **Vertical Motion of a Thrown Ball: Understanding the Equation**When a ball is thrown vertically upward from a platform, its motion can be described using quadratic equations in physics. Consider a ball thrown vertically upward from a height of 6 feet with an initial velocity of 80 feet per second. The position function, which gives the height \( s(t) \) in feet of the ball at any time \( t \) in seconds, is given by the equation:\[ s(t) = 6 + 80t - 16t^2 \]### Explanation of the Equation Components:- \( 6 \): This is the initial height in feet from which the ball is thrown.- \( 80t \): This term represents the initial velocity (80 feet per second) multiplied by time \( t \). This term indicates how the initial upward force affects the height over time.- \( -16t^2 \): This term accounts for the acceleration due to gravity, which is approximately \(-32 \text{ ft/sec}^2\) near Earth's surface. Since the ball is moving upward against gravity, the effect of gravity is subtracted from the height equation.### Understanding the Terms in the Context of Motion:1. **Initial Height (6 feet)**: The ball starts from a platform 6 feet above the ground.2. **Initial Velocity (80 ft/sec)**: The ball's upward movement starts with a force pushing it at a speed of 80 feet per second.3. **Gravity (-16t^2)**: Gravity pulls the ball downward and slows its upward motion until it eventually starts descending.### Practical Analysis:By analyzing this equation, we can predict several things about the ball’s motion:- **Maximum Height**: Occurs when the first derivative of the height function \( s(t) \) with respect to time (velocity) is zero. - **Time to reach Maximum Height**: Calculated using the equation derived from setting the first derivative to zero.- **Total Time of Flight**: The ball's height will be zero when the ball hits the ground, which can be found by setting \( s(t) = 0 \) and solving for \( t \).This equation is an essential part of understanding motion in physics and helps analyze how objects move under forces like gravity.

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A ball is thrown vertically upward from a height of 6 feet with an initial velocity of 80 ft/sec. The height (in feet) of the ball at t seconds is given by s(t) = 6 + 80t - 16t².

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Using the information given in the image, find:

a) the velocity at 2 seconds

b) when it will reach the maximum height

c) what the maximum height is

d) the acceleration at any time (t)

e) when it will hit the ground

f) the velocity at which it will hit the ground

**Vertical Motion of a Thrown Ball: Understanding the Equation**

When a ball is thrown vertically upward from a platform, its motion can be described using quadratic equations in physics.

Consider a ball thrown vertically upward from a height of 6 feet with an initial velocity of 80 feet per second. The position function, which gives the height \( s(t) \) in feet of the ball at any time \( t \) in seconds, is given by the equation:

\[ s(t) = 6 + 80t - 16t^2 \]

### Explanation of the Equation Components:

- \( 6 \): This is the initial height in feet from which the ball is thrown.
- \( 80t \): This term represents the initial velocity (80 feet per second) multiplied by time \( t \). This term indicates how the initial upward force affects the height over time.
- \( -16t^2 \): This term accounts for the acceleration due to gravity, which is approximately \(-32 \text{ ft/sec}^2\) near Earth's surface. Since the ball is moving upward against gravity, the effect of gravity is subtracted from the height equation.

### Understanding the Terms in the Context of Motion:

1. **Initial Height (6 feet)**:
The ball starts from a platform 6 feet above the ground.

2. **Initial Velocity (80 ft/sec)**:
The ball's upward movement starts with a force pushing it at a speed of 80 feet per second.

3. **Gravity (-16t^2)**:
Gravity pulls the ball downward and slows its upward motion until it eventually starts descending.

### Practical Analysis:

By analyzing this equation, we can predict several things about the ball’s motion:

- **Maximum Height**: Occurs when the first derivative of the height function \( s(t) \) with respect to time (velocity) is zero.

- **Time to reach Maximum Height**: Calculated using the equation derived from setting the first derivative to zero.

- **Total Time of Flight**: The ball's height will be zero when the ball hits the ground, which can be found by setting \( s(t) = 0 \) and solving for \( t \).

This equation is an essential part of understanding motion in physics and helps analyze how objects move under forces like gravity.

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