Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 54

A particle leaves its initial position \(x_{0}\) at time \(t=0,\) moving in the positive \(x\) -direction with speed \(v_{0}\) but undergoing acceleration of magnitude \(a\) in the negative \(x\) -direction. Find expressions for (a) the time when it returns to \(x_{0}\) and (b) its speed when it passes that point.

Short Answer

Expert verified
The time when the particle returns to the initial position \(x_{0}\) is \(t = 2v_{0}/a\), and the speed at which it passes that point is \(v = 0\).

Step by step solution

01

Identify Relevant Concepts from Physics

To solve the problem, the equations of motion will be used, one of which states \(x = x_{0} + v_{0}t - 0.5at^2\). This equation comes from the basic definition of acceleration and describes the position of an object given its initial position \(x_{0}\), initial velocity \(v_{0}\), acceleration \(a\) and the time \(t\).
02

Setup and Solve the Equation for Time

The time when the particle returns to its original position means that \(x = x_{0}\). Substituting this information into the motion equation gives: \(x_{0} = x_{0} + v_{0}t - 0.5at^2\). We can simplify this to obtain the equation for the time when the particle returns to its original position \(t = 2v_{0}/a\).
03

Setup and Solve the Equation for Speed

Another equation of motion can give the speed (velocity) of an object given its initial velocity, acceleration and the time, which is \(v = v_{0} - at\). Substituting the time \(t = 2v_{0} / a\) obtained from last step into the speed equation gives: \(v = v_{0} - a(2v_{0}/a)\), simplifying this gives the speed when the particle returns to its original position \(v = 0\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

The standard 26 -mile, 385 -yard marathon dates to \(1908,\) when the Olympic marathon started at Windsor Castle and finished before the Royal Box at London's Olympic Stadium. Today's top marathoners achieve times around 2 hours, 4 minutes for the standard marathon. (a) What's the average speed of a marathon run in this time? (b) Marathons before 1908 were typically about 25 miles. How much longer does the race last today as a result of the extra mile and 385 yards, assuming it's run at the average speed?

An airplane's takeoff speed is \(320 \mathrm{km} / \mathrm{h}\). If its average acceleration is \(2.9 \mathrm{m} / \mathrm{s}^{2},\) how much time is it accelerating down the runway before it lifts off?

You toss a book into your dorm room, just clearing a windowsill \(4.2 \mathrm{m}\) above the ground. (a) If the book leaves your hand \(1.5 \mathrm{m}\) above the ground, how fast must it be going to clear the sill? (b) How long after it leaves your hand will it hit the floor, \(0.87 \mathrm{m}\) below the windowsill?

An object's position is given by \(x=b t+c t^{3},\) where \(b=1.50 \mathrm{m} / \mathrm{s}, \quad c=0.640 \mathrm{m} / \mathrm{s}^{3},\) and \(t\) is time in seconds. To study the limiting process leading to the instantaneous velocity, calculate the object's average velocity over time intervals from (a) \(1.00 \mathrm{s}\) to \(3.00 \mathrm{s},\) (b) \(1.50 \mathrm{s}\) to \(2.50 \mathrm{s},\) and \((\mathrm{c}) 1.95 \mathrm{s}\) to \(2.05 \mathrm{s}\) (d) Find the instantaneous velocity as a function of time by differentiating, and compare its value at 2 s with your average velocities.

Two divers jump from a 3.00 -m platform. One jumps upward at \(1.80 \mathrm{m} / \mathrm{s},\) and the second steps off the platform as the first passes it on the way down. (a) What are their speeds as they hit the water? (b) Which hits the water first and by how much?
See all solutions

Recommended explanations on Physics Textbooks

Waves Physics

Read Explanation

Nuclear Physics

Read Explanation

Quantum Physics

Read Explanation

Measurements

Read Explanation

Electric Charge Field and Potential

Read Explanation

Atoms and Radioactivity

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.

Sign-up for free