/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Flying ball: A ball is tossed up... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

Flying ball: A ball is tossed upward from a tall building, and its upward velocity \(V\), in feet per second, is a function of the time \(t\), in seconds, since the ball was thrown. The formula is \(V=40-32 t\) if we ignore air resistance. The function \(V\) is positive when the ball is rising and negative when the ball is falling. a. Express using functional notation the velocity 1 second after the ball is thrown, and then calculate that value. Is the ball rising or falling then? b. Find the velocity 2 seconds after the ball is thrown. Is the ball rising or falling then? c. What is happening \(1.25\) seconds after the ball is thrown? d. By how much does the velocity change from 1 to 2 seconds after the ball is thrown? From 2 to 3 seconds? From 3 to 4 seconds? Compare the answers to these three questions and explain in practical terms.

Short Answer

Expert verified
1 sec: Rising at 8 ft/s; 2 sec: Falling at -24 ft/s; 1.25 sec: At peak. Velocity changes by -32 ft/s each second, showing constant acceleration.

Step by step solution

01

Express the Velocity for 1 Second

To find the velocity at 1 second, substitute \(t = 1\) in the function \(V(t) = 40 - 32t\). So, \(V(1) = 40 - 32 \times 1\).
02

Calculate the Velocity for 1 Second

Calculate \(V(1) = 40 - 32 \times 1 = 40 - 32 = 8\) feet per second. Since the velocity is positive, the ball is rising 1 second after it's thrown.
03

Express the Velocity for 2 Seconds

To find the velocity at 2 seconds, substitute \(t = 2\) in the same velocity function. Therefore, \(V(2) = 40 - 32 \times 2\).
04

Calculate the Velocity for 2 Seconds

Calculate \(V(2) = 40 - 32 \times 2 = 40 - 64 = -24\) feet per second. Since the velocity is negative, the ball is falling 2 seconds after it's thrown.
05

Determine the State of the Ball at 1.25 Seconds

Substitute \(t = 1.25\) into the velocity function: \(V(1.25) = 40 - 32 \times 1.25\).
06

Calculate the Velocity at 1.25 Seconds

Calculate \(V(1.25) = 40 - 32 \times 1.25 = 40 - 40 = 0\) feet per second. The velocity is zero, meaning the ball is at its highest point at 1.25 seconds.
07

Velocity Change from 1 to 2 Seconds

The change in velocity from 1 second to 2 seconds is \(V(2) - V(1) = -24 - 8 = -32\) feet per second.
08

Velocity Change from 2 to 3 Seconds

Calculate \(V(3)\) by substituting \(t = 3\) into the velocity function: \(V(3) = 40 - 32 \times 3 = -56\). Then find the change \(V(3) - V(2) = -56 - (-24) = -32\) feet per second.
09

Velocity Change from 3 to 4 Seconds

Calculate \(V(4)\) by substituting \(t = 4\) into the velocity function: \(V(4) = 40 - 32 \times 4 = -88\). Then find the change \(V(4) - V(3) = -88 - (-56) = -32\) feet per second.
10

Compare Velocity Changes

The velocity changes by \(-32\) feet per second for every additional second after it's tossed. This consistent change indicates a constant acceleration due to gravity affecting the ball's movement.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Projectile Motion and its Characteristics
When you throw a ball into the air, it follows a path determined by projectile motion. This motion describes the trajectory an object follows when it is thrown, launched, or projected into the air. Several forces influence projectile motion, but in simple physics problems like ours, we often ignore factors like air resistance to focus on gravity. A key feature of projectile motion is the parabolic path that occurs due to these forces.
  • Initial velocity: The velocity at which the ball is thrown upward.
  • Gravity: The force pulling the ball downward after it reaches its highest point.
  • Time of flight: The time the projectile spends moving through the air.
  • Maximum height: The peak the object reaches in its path.
Understanding projectile motion is crucial in many areas of physics, engineering, and everyday occurrences like throwing a ball during a game.
The Role of Gravity in Motion
Gravity is a fundamental force impacting projectile motion, and it's the reason objects fall back to the ground after they're thrown upward. On Earth, gravity accelerates objects downward at approximately 32 feet per second squared. This constant acceleration is what affects the velocity of the ball in our problem. As the ball rises, gravity slows it down until it reaches its peak height, where the velocity becomes zero.

After reaching this point, gravity causes the ball to accelerate downward, increasing its speed on its way back to the ground. In the velocity function given by the formula, you can observe gravity's constant effect as it alters the velocity by 32 feet per second in each subsequent second. This knowledge helps us predict the object鈥檚 behavior at different points in time.
Algebraic Modeling in Physics
Algebraic modeling enables us to express real-world situations, like projectile motion, using mathematical equations. In our exercise, the velocity function is expressed as an algebraic equation: \( V(t) = 40 - 32t \). This equation models the upward and then downward motion of the ball over time.
  • Starting velocity: The 40 in the equation represents initial upward velocity when \( t = 0 \).
  • Gravity's effect: The term \(-32t\) indicates the decrease in velocity due to gravity over time.
  • Variables: \( t \) acts as a variable representing the time.
By using algebraic modeling, we can plug in different time values into the equation to determine the ball's velocity at any given second. It's a way to predict future or past behavior of systems without direct experimentation.
Understanding Function Notation
Function notation is a method of representing relationships in mathematics, notably used to describe how variables depend on each other. In the context of projectile motion, function notation is used to express formulas for velocity or position based on time. Consider the velocity function \( V(t) \). Here, \( V \) is the function that describes the velocity at any time \( t \).

To use function notation:
  • Select a specific time, for instance, \( t = 1 \), and substitute it into the function to find the corresponding velocity, \( V(1) = 8 \).
  • Repeat this for any other desired time to see how velocity changes over the course of the motion.
  • Analyze the results to determine the state of the object's motion, such as rising or falling.
Function notation allows for an organized way to represent complex relationships in physics, helping us better understand and communicate the behavior of moving objects.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A population of deer: When a breeding group of animals is introduced into a restricted area such as a wildlife reserve, the population can be expected to grow rapidly at first but to level out when the population grows to near the maximum that the environment can support. Such growth is known as logistic population growth, and ecologists sometimes use a formula to describe it. The number \(N\) of deer present at time \(t\) (measured in years since the herd was introduced) on a certain wildlife reserve has been determined by ecologists to be given by the function $$ N=\frac{12.36}{0.03+0.55^{t}} $$ a. How many deer were initially on the reserve? b. Calculate \(N(10)\) and explain the meaning of the number you have calculated. c. Express the number of deer present after 15 years using functional notation, and then calculate it. d. How much increase in the deer population do you expect from the 10 th to the 15 th year?

Growth in weight: The following table gives, for a certain man, his weight \(W=W(t)\) in pounds at age \(t\) in years. $$ \begin{array}{|c|c|} \hline t=\text { Age } \\ \text { (years) } & \begin{array}{c} W=\text { Weight } \\ \text { (pounds) } \end{array} \\ \hline 4 & 36 \\ \hline 8 & 54 \\ \hline 12 & 81 \\ \hline 16 & 128 \\ \hline 20 & 156 \\ \hline 24 & 163 \\ \hline \end{array} $$ a. Make a table showing, for each of the 4-year periods, the average yearly rate of change in \(W\). b. Describe in general terms how the man's gain in weight varied over time. During which 4 -year period did the man gain the most in weight? c. Estimate how much the man weighed at age 30 . d. Use the average rate of change to estimate how much he weighed at birth. Is your answer reasonable?

Sketching a graph with given concavity: a. Sketch a graph that is always decreasing but starts out concave down and then changes to concave up. There should be a point of inflection in your picture. Mark and label it. b. Sketch a graph that is always decreasing but starts out concave up and then changes to concave down. There should be a point of inflection in your picture. Mark and label it.

What if interest is compounded more often than monthly? Some lending institutions compound interest daily or even continuously. (The term continuous compounding is used when interest is being added as often as possible - that is, at each instant in time.) The point of this exercise is to show that, for most consumer loans, the answer you get with monthly compounding is very close to the right answer, even if the lending institution compounds more often. In part 1 of Example 1.2, we showed that if you borrow \(\$ 7800\) from an institution that compounds monthly at a monthly interest rate of \(0.67 \%\) (for an APR of \(8.04 \%\) ), then in order to pay off the note in 48 months, you have to make a monthly payment of \(\$ 190.57\). a. Would you expect your monthly payment to be higher or lower if interest were compounded daily rather than monthly? Explain why. b. Which would you expect to result in a larger monthly payment, daily compounding or continuous compounding? Explain your reasoning. c. When interest is compounded continuously, you can calculate your monthly payment \(M=\) \(M(P, r, t)\), in dollars, for a loan of \(P\) dollars to be paid off over \(t\) months using $$ M=\frac{P\left(e^{r}-1\right)}{1-e^{-r t}}, $$ where \(r=\frac{A P R}{12}\) if the APR is written in decimal form. Use this formula to calculate the monthly payment on a loan of \(\$ 7800\) to be paid off over 48 months with an APR of \(8.04 \%\). How does this answer compare with the result in Example 1.2?

A rental: A rental car agency charges \(\$ 49.00\) per day and 25 cents per mile. a. Calculate the rental charge if you rent a car for 2 days and drive 100 miles. b. Use a formula to express the cost of renting a car as a function of the number of days you keep it and the number of miles you drive. Identify the function and each variable you use, and state the units. c. It is about 250 miles from Dallas to Austin. Use functional notation to express the cost to rent a car in Dallas, drive it to Austin, and return it in Dallas 1 week later. Use the formula from part b to calculate the cost.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.