Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 491

A body initially at rest undergoes one dimensional motion with constant acceleration. The power delivered to it at time \(\mathrm{t}\) is proportional to..... (A) \(\mathrm{t}^{1 / 2}\) (B) \(t\) (C) \(\mathrm{t}^{3 / 2}\) (D) \(\mathrm{t}^{2}\)

Short Answer

Expert verified
The power delivered to the body at time \(t\) is proportional to \(t\).

Step by step solution

01

Identify the known values

Since the body is initially at rest, it is given that Initial velocity, \(v_0 = 0\). Also, it undergoes constant acceleration, so there exists a constant acceleration \(a\). We need to find the power delivered to the body at time \(t\).
02

Find the formula for work done (w)

The work-energy theorem states that the work done (W) on the body equals the change in its kinetic energy. \[W = \frac{1}{2}m(v^2 - v_0^2)\] Since \(v_0 = 0\), we get: \[W = \frac{1}{2}mv^2\]
03

Express velocity (v) in terms of acceleration (a) and time (t)

From the equation of motion with constant acceleration, \(v = v_0 + at\) Since initial velocity \(v_0 = 0\), we have: \(v = at\)
04

Substitute the expression for velocity in the work done formula

Replace \(v\) with \(at\) in the work done formula: \[W = \frac{1}{2}m(at)^2 = \frac{1}{2}ma^2t^2\]
05

Calculate the power (P) formula

The power (P) is the rate at which work is done, so we need to find the derivative of the work with respect to time (t). \[P = \frac{dW}{dt}\] Differentiating the work formula with respect to time: \[P = \frac{d}{dt}\left(\frac{1}{2}ma^2t^2\right) = ma^2t\]
06

Find the proportion of power to time

Consider the power formula \(P = ma^2t\). We need to determine which of the given options best represents the proportionality between power and time. Comparing the formula with each option, we get that \(P \propto t^{1}\) Therefore, the correct option is: (B) \(t\)

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

An electric motor develops \(5 \mathrm{KW}\) of power. How much time will it take to lift a water of mass \(100 \mathrm{~kg}\) to a height of $20 \mathrm{~m}\( ? \)\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$ (A) \(4 \mathrm{sec}\) (B) \(5 \mathrm{sec}\) (C) \(8 \mathrm{sec}\) (D) \(10 \mathrm{sec}\)

Two bodies of masses \(m_{1}\) and \(m_{2}\) have equal kinetic energies. If \(P_{1}\) and \(P_{2}\) are their respective momentum, what is ratio of \(\mathrm{P}_{2}: \mathrm{P}_{1}\) ? (A) \(\mathrm{m}_{1}: \mathrm{m}_{2}\) (B) \(\sqrt{\mathrm{m}}_{2} / \sqrt{\mathrm{m}_{1}}\) (C) \(\sqrt{m_{1}}: \sqrt{m_{2}}\) (D) \(\mathrm{m}_{1}^{2}: \mathrm{m}_{2}^{2}\)

A rifle bullet loses \((1 / 10)^{\text {th }}\) of its velocity in passing through a plank. The least number of such planks required just to stop the bullet is (A) 5 (B) 10 (C) 11 (D) 20

Natural length of a spring is \(60 \mathrm{~cm}\), and its spring constant is \(2000 \mathrm{~N} / \mathrm{m}\). A mass of \(20 \mathrm{~kg}\) is hung from it. The extension produced in the spring is..... $\left(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^{2}\right)$ (A) \(4.9 \mathrm{~cm}\) (B) \(0.49 \mathrm{~cm}\) (C) \(9.8 \mathrm{~cm}\) (D) \(0.98 \mathrm{~cm}\)

A neutron having mass of \(1.67 \times 10^{-27} \mathrm{~kg}\) and moving at \(10^{8} \mathrm{~m} / \mathrm{s}\) collides with a deuteron at rest and sticks to it. If the mass of the deuteron is \(3.34 \times 10^{-27} \mathrm{~kg}\) then the speed of the combination is (A) \(3.33 \times 10^{7} \mathrm{~m} / \mathrm{s}\) (B) \(3 \times 10^{5} \mathrm{~m} / \mathrm{s}\) (C) \(33.3 \times 10^{7} \mathrm{~m} / \mathrm{s}\) (D) \(2.98 \times 10^{5} \mathrm{~m} / \mathrm{s}\)
See all solutions

Recommended explanations on English Textbooks

Essay Plan

Read Explanation

Blog

Read Explanation

Global English

Read Explanation

Free Response Essay

Read Explanation

Pragmatics

Read Explanation

Listening and Speaking

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