**Chapter 2 - The Derivative Function**### Problem 1:The distance a person can walk in time \( t \) is given by \( d = \sqrt{t} \), where \( d \) is measured in miles and \( t \) is measured in hours.a) **Find the average velocity** from \( t = 1 \) to \( t = 4 \) hours.b) **Estimate the person's velocity** at \( t = 4 \) hours.### Problem 2:A ball is dropped from a tall building and \( d \) is the number of feet it falls in \( t \) seconds. The following table shows some of the values of \( t \) and \( d \).\[\begin{array}{|c|c|c|c|c|c|}\hlinet \, (\text{sec}) & 0 & 1 & 2 & 3 & 4 & 5 \\\hlined \, (\text{ft}) & 0 & 15.07 & 57.90 & 127.03 & 201.76 & 333.00 \\\hline\end{array}\]a) **Plot the six data points.** Label the axes and indicate the units that are being used.b) **Explain why the slope of any secant line on the graph is a measure of velocity.**c) **Make three estimates** of the velocity of the falling ball at \( t = 2 \) seconds.### Explanation:- **Secant Line Slope**: The slope of a secant line between two points on a distance-time graph represents the average velocity over that time interval.- **Velocity Estimation**: By choosing points around \( t = 2 \) seconds, you can approximate the instantaneous velocity as the slope of the tangent at that point.This exercise explores concepts of average vs. instantaneous velocity using basic calculus and physics principles.

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Description: **Chapter 2 - The Derivative Function**### Problem 1:The distance a person can walk in time \( t \) is given by \( d = \sqrt{t} \), where \( d \) is measured in miles and \( t \) is measured in hours.a) **Find the average velocity** from \( t = 1 \)

Distance as a function of time is given by; d=t

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The distance a person can walk in time t is given by d = vt where d is measured in miles and t is measured in hours. a) Find the average velocity from t = 1 to t = 4 hours. b) Estimate the person's velocity at t = 4 hours.

The distance a person can walk in time t is given by d = vt where d is measured in miles and t is measured in hours. a) Find the average velocity from t = 1 to t = 4 hours. b) Estimate the person's velocity at t = 4 hours.

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Displacement, Velocity and Acceleration

In classical mechanics, kinematics deals with the motion of a particle. It deals only with the position, velocity, acceleration, and displacement of a particle. It has no concern about the source of motion.

Linear Displacement

The term "displacement" refers to when something shifts away from its original "location," and "linear" refers to a straight line. As a result, “Linear Displacement” can be described as the movement of an object in a straight line along a single axis, for example, from side to side or up and down. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Non-contact sensors such as LVDTs and other linear location sensors can calculate linear displacement. Linear displacement is usually measured in millimeters or inches and may be positive or negative.

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**Chapter 2 - The Derivative Function**

### Problem 1:
The distance a person can walk in time \( t \) is given by \( d = \sqrt{t} \), where \( d \) is measured in miles and \( t \) is measured in hours.

a) **Find the average velocity** from \( t = 1 \) to \( t = 4 \) hours.

b) **Estimate the person's velocity** at \( t = 4 \) hours.

### Problem 2:
A ball is dropped from a tall building and \( d \) is the number of feet it falls in \( t \) seconds. The following table shows some of the values of \( t \) and \( d \).

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
t \, (\text{sec}) & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
d \, (\text{ft}) & 0 & 15.07 & 57.90 & 127.03 & 201.76 & 333.00 \\
\hline
\end{array}
\]

a) **Plot the six data points.** Label the axes and indicate the units that are being used.

b) **Explain why the slope of any secant line on the graph is a measure of velocity.**

c) **Make three estimates** of the velocity of the falling ball at \( t = 2 \) seconds.

### Explanation:
- **Secant Line Slope**: The slope of a secant line between two points on a distance-time graph represents the average velocity over that time interval.
- **Velocity Estimation**: By choosing points around \( t = 2 \) seconds, you can approximate the instantaneous velocity as the slope of the tangent at that point.

This exercise explores concepts of average vs. instantaneous velocity using basic calculus and physics principles.

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