Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 43

A particle moves so that its position as a function of time in MKS unit is \(\vec{r}=\hat{i}+4 t^{2} \hat{j}+t \hat{k}\). The trajectory of the particle is - (1) parabola (2) ellipse (3) circle (4) None of these

Short Answer

Expert verified
The trajectory of the particle is a parabola.

Step by step solution

01

Analyze the Position Vector

The position vector \(\vec{r}\) is given by \(\vec{r} = \hat{i} + 4t^2 \hat{j} + t \hat{k}\). This means the x-component is constant at 1, the y-component varies as \(4t^2\), and the z-component varies linearly with \(t\).
02

Identify the Motion in the xy-plane

In the xy-plane, the x-component is constant (x = 1), and y varies as \(4t^2\). This corresponds to the parabolic equation \(y = 4t^2\).
03

Identify the Motion in the xz-plane

In the xz-plane, the x-component is constant (x = 1), and z varies linearly with \(t\). Therefore, \(z = t\).
04

Conclusion about the Trajectory

Combining the analyses, the particle follows a parabolic path in the xy-plane. Therefore, the trajectory of the particle is a parabola.

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!

Key Concepts

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

Position Vector
To understand the movement of a particle, it's crucial to examine its position vector. The position vector \(\backslash{}vec{r} = \backslash{}hat{i} + 4t^2 \backslash{}hat{j} + t \backslash{}hat{k}\) represents the location of the particle in a 3D space at any given time. Here, components \(\backslash{}hat{i}, \backslash{}hat{j},\) and \(\backslash{}hat{k}\) denote the x, y, and z directions, respectively.
In this function:
  • The x-component is 1 (constant)
  • The y-component changes with time squared \((4t^2)\)
  • The z-component changes linearly with time \(t\)

This information helps us visualize the particle's motion through space as time progresses.
Parabolic Trajectory
A parabolic trajectory is a common path in physics, usually resulting from uniform acceleration in one direction and constant motion in another.
Looking at the xy-plane:
  • The x-component remains constant at x = 1
  • Y-component varies as \(4t^2\)

This gives us the equation for a parabola, y = 4t^2, indicating that in the xy-plane the particle traces a parabolic path.
Parabolic motion is often seen in projectile motion where the object accelerates due to gravity (in the y-direction) while maintaining constant horizontal speed.
Motion in Planes
Motion in planes can be analyzed by observing the movement in two-dimensional spaces formed by pairs of axes.

In the xy-plane:
  • X remains constant at x = 1
  • Y varies as y = 4t^2

    • In the xz-plane:
    • X remains constant at x = 1
    • Z varies linearly as z = t

    Combining these observations helps us understand the overall motion of the particle.
    Here, the particle’s trajectory in 3D space is seen as a combination of parabolic motion in the xy-plane and linear motion in the xz-plane, reinforcing the conclusion of a parabolic trajectory.

    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 ring of mass \(\mathrm{m}\) and radius \(\mathrm{R}\) is rolling without sliding on a flat fixed horizontal surface, with a constant angular velocity. The angle between the velocity and acceleration vectors of a point \(\mathrm{P}\) on the rim of the ring at the same horizontal level as the centre of the ring may be - (1) zero (2) \(90^{\circ}\) (3) \(135^{\circ}\) (4) \(\tan ^{-1} 1 / 2\)

    Set of equations \(x^{2}-b_{n} x+\underbrace{111 \ldots . .1}_{n-\text { times }}=n, \quad n=2,3,4, \ldots \ldots, 9\) and \(\mathrm{b}_{\mathrm{n}} \in \mathrm{N}\), have integer roots. Which of the following are CORRECT ? (1) Common root of these equations is \(9 .\) (2) No three of these equations have a common root. (3) A root of an equation with \(\mathrm{n}=4\) can be \(123 .\) (4) A root of an equation with \(\mathrm{n}=4\) is 1234 .

    An office employs thirty people. Five of them speak Sanskrit, Hindi, English. Nine speak Sanskrit, English. Twenty speak Hindi of which twelve also speak Sanskrit. Eighteen speak English. No one speak only Sanskrit. An employee selected at random is known to speak either Hindi or English, the probability that employee speaks only two of three languages is (1) \(14 / 15\) (2) \(7 / 15\) (3) \(1 / 6\) (4) \(5 / 6\)

    \(\mathrm{S}=\sum_{\mathrm{r}=1}^{7} \tan ^{2} \frac{\mathrm{r} \pi}{16}\), then \(\mathrm{S}\) is equal to (1) 44 (2) 40 (3) 34 (4) 35

    A piece of string is cut into two pieces. The point at which the string is cut was chosen at random. What is the probability that the longer piece is at least three times as long as the shorter piece? (1) \(1 / 4\) (2) \(1 / 3\) (3) \(2 / 5\) (4) \(1 / 2\)
    See all solutions

    Recommended explanations on English Textbooks

    Lexis and Semantics Summary

    Read Explanation

    Linguistic Terms

    Read Explanation

    Graphology

    Read Explanation

    Prosody

    Read Explanation

    Textual Analysis

    Read Explanation

    Lexis and Semantics

    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