/*! 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 33 Give an example of: A function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 33

Give an example of: A function giving the position of a particle that has the same speed at \(t=-1\) and \(t=1\) but different velocities.

Short Answer

Expert verified
The function \( f(t) = t^3 - 3t \) has the same speed at \( t=-1 \) and \( t=1 \) but different velocities.

Step by step solution

01

Define the Function

We need a function that represents the position of a particle over time. Let's consider the function \( f(t) = t^3 - 3t \). This is a cubic function that will give us both position and allows us to calculate speed and velocity at given times.
02

Calculate the Velocity

Velocity is the derivative of the position function with respect to time. Compute the derivative of \( f(t) \): \ v(t) = \frac{d}{dt}(t^3 - 3t) = 3t^2 - 3. \ This function represents the velocity at any time \( t \).
03

Evaluate Velocity at \( t = -1 \) and \( t = 1 \)

Substitute \( t = -1 \) into the velocity function: \ v(-1) = 3(-1)^2 - 3 = 3 \times 1 - 3 = 0. \ Substitute \( t = 1 \) into the velocity function: \ v(1) = 3(1)^2 - 3 = 3 \times 1 - 3 = 0. \ The velocities at \( t = -1 \) and \( t = 1 \) are both 0, indicating that the particle stops momentarily at these points, pointing to a direction change.
04

Calculate the Speed

Speed is the absolute value of velocity. Calculate the speed at \( t = -1 \) and \( t = 1 \): \ \text{Speed at } t = -1 \text{ is } |v(-1)| = |0| = 0. \ \text{Speed at } t = 1 \text{ is } |v(1)| = |0| = 0. \ Both times, the speed is 0.
05

Conclusion

The function \( f(t) = t^3 - 3t \) satisfies the conditions given in the problem. It has the same speed (0) at \( t = -1 \) and \( t = 1 \) but changes direction (indicating different velocity directions) at these times.

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.

Position Function
In calculus, the position function is a fundamental concept used to describe the location of a particle at any given time. For our example, the position of a particle is given by the function \( f(t) = t^3 - 3t \). This function tells us how the particle's position changes over time. By substituting any value of \( t \) into this function, you can find the particle's position at that particular time.
  • For instance, if you plug in \( t = 0 \), the particle's position is \( f(0) = 0^3 - 3 \times 0 = 0 \)
  • The function is a continuous cubic equation, which means the graph is a smooth curve, indicating continuous motion over time
The position function is crucial because it is the starting point for finding other related concepts, like velocity and speed, through its derivatives.
Derivative
The derivative is a tool in calculus that helps us understand how a function is changing at any given point. When dealing with the position function, the derivative tells us the velocity of the particle, which is the rate of change of position.In our example, we take the derivative of the position function \( f(t) = t^3 - 3t \) to find the velocity function:\[v(t) = \frac{d}{dt}(t^3 - 3t) = 3t^2 - 3\]
  • This derivative indicates how fast and in which direction the particle is moving over time
  • The operation of differentiation reduces the power of each contributing factor in the polynomial
The derivative is fundamental for determining how the position of a particle varies as time progresses, making it possible to deduce further properties like velocity and acceleration.
Velocity
Velocity is a vector quantity that represents the rate of change of position with respect to time. In other words, velocity describes how quickly and in what direction a particle is moving.For the position function \( f(t) = t^3 - 3t \), we found the velocity function to be \( v(t) = 3t^2 - 3 \).
  • At \( t = -1 \), the velocity is \( v(-1) = 3(-1)^2 - 3 = 0 \)
  • Similarly, at \( t = 1 \), the velocity is \( v(1) = 3(1)^2 - 3 = 0 \)
These evaluations show that the particle's velocity is zero at both \( t = -1 \) and \( t = 1 \). This indicates that the particle has come to a temporary stop at those times, possibly reversing its direction of motion.
Speed
Speed is the magnitude of velocity and is always a non-negative quantity. Unlike velocity, speed does not give the direction of movement, only how fast the particle is moving.In this example, speed is calculated as the absolute value of the velocity:
  • At \( t = -1 \), the speed is \( |v(-1)| = |0| = 0 \)
  • At \( t = 1 \), the speed is also \( |v(1)| = |0| = 0 \)
Speed emphasizes how quickly the particle is traveling at a specific time, regardless of the direction, providing a simpler interpretation compared to velocity. In this case, since the speed is zero at both times, the particle is momentarily stationary.

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

Let \(P(t)\) represent the price of a share of stock of a corporation at time \(t .\) What does each of the following statements tell us about the signs of the first and second derivatives of \(P(t) ?\) (a) "The price of the stock is rising faster and faster." (b) "The price of the stock is close to bottoming out."

Chlorofluorocarbons (CFCs) were used as propellants in spray cans until their build up in the atmosphere started destroying the ozone, which protects us from ultraviolet rays. since the 1987 Montreal Protocol (an agreement to curb CFCs), the CFCs in the atmosphere above the US have been reduced from a high of 3200 parts per trillion (ppt) in 1994 to 2750 ppt in \(2010 .^{15}\) The reduction has been approximately linear. Let \(C(t)\) be the concentration of CFCs in ppt in year \(t\) (a) Find \(C(1994)\) and \(C(2010)\) (b) Estimate \(C^{\prime}(1994)\) and \(C^{\prime}(2010)\) (c) Assuming \(C(t)\) is linear, find a formula for \(C(t)\) (d) When is \(C(t)\) expected to reach 1850 ppt, the level before CFCs were introduced? (e) If you were told that in the future, \(C(t)\) would not be exactly linear, and that \(C^{\prime \prime}(t)>0,\) would your answer to part (d) be too early or too late?

If \(g(v)\) is the fuel efficiency, in miles per gallon, of a car going at \(v\) miles per hour, what are the units of \(g^{\prime}(90) ?\) What is the practical meaning of the statement \(g^{\prime}(55)=-0.54 ?\)

Are the statements true or false? Give an explanation for your answer. If \(f^{\prime}(x)\) is increasing, then \(f(x)\) is also increasing.

Give an explanation for your answer.If \(f(x)\) is concave up, then \(f^{\prime}(a)<(f(b)-f(a)) /(b-a)\) for \(a
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.