/*! 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 20 Assume a canister in a straight ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 20

Assume a canister in a straight tube moves with a constant acceleration of \(-4.00 \mathrm{~m} / \mathrm{s}^{2}\) and has a velocity of \(13.0 \mathrm{~m} / \mathrm{s}\) at \(\ell=0 .\) (a) What is its velocity at \(t=1.00 \mathrm{~s} ?\) (c) At \(t=2.50\) s? (b) At \(t=2.00 \mathrm{~s}^{2}\) ( (d) At \(t=4.00 \mathrm{s?}\) (e) Describe the shape of the canister's velocity versus time graph. (f) What two things must be known at a given time to predict the canister's velocity at any later time?

Short Answer

Expert verified
The velocities of the canister at \(t=1.00s, t=2.50s, t=2.00s\), and \(t=4.00s\) are calculated using the equation of motion. The velocity of canister gradually decreases due to negative acceleration. The graph representing velocity vs. time is a straight line sloping downwards. The initial velocity and the acceleration must be known to predict the canister's velocity.

Step by step solution

01

Calculate Velocity at \(t=1.00\) s

Use the first equation of motion \(v_f = v_i + a \cdot t\) where \(v_f\) is the final velocity, \(v_i\) is the initial velocity, \(a\) is the acceleration and \(t\) is the time. Substitute \(v_i = 13.0 \mathrm{~m/s}\), \(a = -4.00 \mathrm{~m/s^2}\), and \(t = 1.00 s\) to find \(v_f\).
02

Calculate Velocity at \(t=2.50\) s

Use the first equation of motion with \(v_i = 13.0 \mathrm{~m/s}\), \(a = -4.00 \mathrm{~m/s^2}\), and \(t = 2.50 s\) to find the new \(v_f\).
03

Calculate Velocity at \(t=2.00 s\)

Again, use the first equation of motion with \(t = 2.00 s\) to find the velocity at this time.
04

Calculate Velocity at \(t=4.00 s\)

Repeat prior steps for time \(t = 4.00 s\) to find the velocity at this moment.
05

Describe the Graph

As the acceleration is constant and negative, the velocity vs. time graph will be a straight line that decreases over time, starting from the initial velocity and sloping downwards from there.
06

Predicting Future Velocity

To predict the canister's velocity at any later time, we need to know both the initial velocity at a known time and the constant acceleration.

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.

Velocity
Velocity is a fundamental concept in kinematics that describes the speed of an object in a specific direction. It is different from speed, which only measures how fast an object is moving without regard to direction. Velocity is a vector quantity, meaning it has both magnitude and direction.
  • Magnitude: How fast the object is moving.
  • Direction: The path the object is taking.
In the context of the canister moving in a straight tube, if the velocity at the initial point (\(t=0\)) is given as \(13.0 \,\mathrm{m/s}\), it indicates how fast and in which direction the canister is moving at that very moment. Remembering that velocity can change over time helps us understand how forces (like acceleration) influence motion.
Acceleration
Acceleration is the rate at which an object's velocity changes over time. It tells us how fast the velocity of an object is increasing or decreasing. Like velocity, acceleration is also a vector quantity, involving magnitude and direction.
  • Positive Acceleration: When speed increases in a particular direction.
  • Negative Acceleration (Deceleration): When speed decreases, like in the case of the canister, which has an acceleration of \(-4.00 \, \mathrm{m/s^2}\).
In this exercise, the constant negative acceleration indicates that the velocity of the canister is decreasing over time. This happens at a steady rate due to the acceleration being constant, resulting in uniform changes in velocity.
Equations of Motion
The equations of motion are mathematical formulas used to predict an object's future state in terms of velocity, position, and acceleration. These equations are crucial in kinematics for calculating how different variables are interrelated over specific time intervals.
  • First Equation of Motion: \(v_f = v_i + a \cdot t\)
This equation helps us compute final velocity \(v_f\) when initial velocity \(v_i\), acceleration \(a\), and time \(t\) are known. In the exercise context, using given values, substituting them into the first equation of motion allows for calculating precise future velocities. This clear methodology is powerful, enabling predictive insights on an object's movement.
Velocity-Time Graph
A velocity-time graph visually represents how an object's velocity changes over time. We'll draw it with the initial velocity on the y-axis at time zero and observe how it progresses.
  • With constant negative acceleration, as present in this exercise with \(-4.00 \, \mathrm{m/s^2}\), the graph appears as a straight, downward-sloping line.
  • The slope of this line equals the acceleration.
By examining the graph, it's possible to identify key patterns in motion, such as when the object is speeding up, slowing down, or moving at a constant velocity. Understanding the shape and slope provides clear insights into motion dynamics, making it easier to predict future states based on current information.

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 tennis player tosses a tennis ball straight up and then catches it after \(2.00 \mathrm{~s}\) at the same height as the point of release. (a) What is the acceleration of the ball while it is in flight? (b) What is the velocity of the ball when it reaches its maximum height? Find (c) the initial velocity of the ball and (d) the maximum height it reaches.

Traumatic brain injury such as concussion results when the head undergoes a very large acceleration. Generally, an acceleration less than \(800 \mathrm{~m} / \mathrm{s}^{2}\) lasting for any length of time will not cause injury, whereas an acceleration greater than \(1000 \mathrm{~m} / \mathrm{s}^{2}\) lasting for at least \(1 \mathrm{~ms}\) will cause injury. Suppose a small child rolls off a bed that is \(0.40 \mathrm{~m}\) above the floor. If the floor is hardwood, the child's head is brought to rest in approximately \(2.0 \mathrm{~mm}\). If the floor is carpeted, this stopping distance is increased Io about \(1.0 \mathrm{~cm}\). Calculate the magnitude and duration of the deceleration in both eases, to determine the risk of injury. Assume the child remains horizontal during the fall to the floor. Note that a more complicated fall could result in a head velocity greater or less than the speed you calculate.

A car traveling east at \(40.0 \mathrm{~m} / \mathrm{s}\) passes a trooper hiding at the roadside. The driver uniformly reduces his speed to \(25.0 \mathrm{~m} / \mathrm{s}\) in \(3.50 \mathrm{~s}\). (a) What is the magnitude and direction of the car's acceleration as it slows down? (b) How far does the car travel in the \(3.5-\mathrm{s}\) time period?

A speedboat increases its speed uniformly from \(v_{t}=\) \(20.0 \mathrm{~m} / \mathrm{s}\) to \(v_{j}=30.0 \mathrm{~m} / \mathrm{s}\) in a distance of \(2.00 \times 10^{2} \mathrm{~m}\) (a) Draw a coordinate system for this situation and label the relevant quantities, including vectors. (b) For the given information, what single equation is most appropriate for finding the acceleration? (c) Solve the equation selected in part (b) symbolically for the boat's acceleration in terms of \(v_{f}, v_{a}\), and \(\Delta x\). (d) Substitute given values, obtaining that acceleration. (e) Find the time it takes the boat to travel the given distance.

A ball is thrown upward from the ground with an initial speed of \(25 \mathrm{~m} / \mathrm{s}\); at the same instant, another ball is dropped from a building \(15 \mathrm{~m}\) high. After how long will the balls be at the same height?
See all solutions

Recommended explanations on Physics Textbooks

Quantum Physics

Read Explanation

Translational Dynamics

Read Explanation

Rotational Dynamics

Read Explanation

Electricity and Magnetism

Read Explanation

Magnetism

Read Explanation

Fluids

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.