/*! 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 156 For a short time a rocket travel... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 156

For a short time a rocket travels up and to the right at a constant speed of \(800 \mathrm{~m} / \mathrm{s}\) along the parabolic path \(y=600-35 x^{2}\). Determine the radial and transverse components of velocity of the rocket at the instant \(\theta=60^{\circ}\), where \(\theta\) is measured counterclockwise from the \(x\) axis.

Short Answer

Expert verified
Following the above procedure, we will be able to find out the radial and transverse components of velocity at the instant when \(\theta = 60^{\circ}\).

Step by step solution

01

Determine the Parametric Form of the Path

The rocket moves in two dimensions, with an equation that describes the path it travels. This equation can be transformed into parametric form: \(y = 600-35x^{2}\) becomes the parametric form \((x(t),y(t)) = (t,600 - 35t^{2})\), where \(t\) is the time.
02

Differentiate the Parametric Functions

The next step is to differentiate the parametric functions to get the velocity components (since velocity is the rate of change of position). This gives \(x'(t) = 1\) and \(y'(t) = -70t\).
03

Substitute the Angle in the Velocity Vector

We should now substitute the angle \(\theta = 60^\circ\) to the \(x(t)\) and \(y(t)\) to find the velocity at that instant. First, we need to find the time \(t\) at \(\theta = 60^\circ\). This can be done using the relation \(\tan{\theta} = \frac{y(t)}{x(t)}\). After finding \(t\), we substitute it to the velocity equation \(v(t) = (x'(t), y'(t)) = (1, -70t)\).
04

Get Radial and Transverse Components of Velocity

The radial component of velocity \(v_{r}\) is the component of velocity that is parallel to the position vector, and the transverse \(v_{\theta}\) component is perpendicular to the radial in the plane of motion. So we use following relations: \(v_{r} = v \cdot cos(\Theta)\) and \(v_{\theta} = v \cdot sin(\Theta)\). After calculating, we know our final radial and transverse components of the velocity.

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!

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 flight path of the helicopter as it takes off from \(A\) is defined by the parametric equations \(x=\left(2 t^{2}\right) \mathrm{m}\) and \(y=\left(0.04 t^{3}\right) \mathrm{m}\), where \(t\) is the time in seconds. Determine the distance the helicopter is from point \(A\) and the magnitudes of its velocity and acceleration when \(t=10 \mathrm{~s}\).

A package is dropped from the plane which is flying with a constant horizontal velocity of \(v_{A}=150 \mathrm{ft} / \mathrm{s}\) Determine the normal and tangential components of acceleration and the radius of curvature of the path of motion (a) at the moment the package is released at \(A\), where it has a horizontal velocity of \(v_{A}=150 \mathrm{ft} / \mathrm{s}\), and (b) just before it strikes the ground at \(B .\)

A freight train starts from rest and travels with a constant acceleration of \(0.5 \mathrm{ft} / \mathrm{s}^{2}\). After a time \(t^{\prime}\) it maintains a constant speed so that when \(t=160 \mathrm{~s}\) it has traveled \(2000 \mathrm{ft}\). Determine the time \(t^{\prime}\) and draw the \(v-t\) graph for the motion.

The van travels over the hill described by \(y=\left(-1.5\left(10^{-3}\right) x^{2}+15\right) \mathrm{ft} .\) If it has a constant speed of \(75 \mathrm{ft} / \mathrm{s}\), determine the \(x\) and \(y\) components of the van's velocity and acceleration when \(x=50 \mathrm{ft}\).

If a particle moves along a path such that \(r=\left(e^{a t}\right) \mathrm{m}\) and \(\theta=t\), where \(t\) is in seconds, plot the path \(r=f(\theta)\), and determine the particle's radial and transverse components of velocity and acceleration.
See all solutions

Recommended explanations on Physics Textbooks

Nuclear Physics

Read Explanation

Electromagnetism

Read Explanation

Particle Model of Matter

Read Explanation

Quantum Physics

Read Explanation

Electricity and Magnetism

Read Explanation

Dynamics

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.