/*! 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 8 Is there any point on a projecti... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 8

Is there any point on a projectile's trajectory where velocity and acceleration are perpendicular?

Short Answer

Expert verified
Yes, at the maximum height of a projectile's trajectory, the vectors of velocity and acceleration are perpendicular because at the maximum height, the vertical component of velocity is zero and acceleration due to gravity acts downward. This is proven when the dot product of the velocity and acceleration vectors equals zero, which happens at time \( t = v_{y0}/g \)

Step by step solution

01

- Understand the Concept of Projectile

A projectile is any object thrown into space upon which the only acting force is gravity. In case of a projectile, acceleration is due to gravity and thus it is constant and acts downward. Mathematically it is represented as \( \vec{a} = -g \hat{j} \) where \(\hat{j}\) is the direction (downwards) and \( g \) is the acceleration due to gravity.
02

- Determine the Velocity Vector

The velocity of a projectile can be split into two components - horizontal (vx) and vertical (vy). The horizontal component remains constant throughout the trajectory, while the vertical component continually changes due to acceleration from gravity. Mathematically, velocity at time \( t \) can be represented as \( \vec{v} = v_x\hat{i} + (v_{y0} - gt)\hat{j} \) here \( \hat{i} \) represents the horizontal direction (positive), \( v_x \) the constant initial horizontal component of velocity, \( v_{y0} \) the initial vertical component of velocity and \( t \) is the time duration.
03

- Determine the Conditions for Perpendicularity

Two vectors are said to be perpendicular if their dot product is zero. The dot product of acceleration vector \( \vec{a} = -g \hat{j} \) and the velocity vector \( \vec{v} = v_x\hat{i} + (v_{y0} - gt)\hat{j} \) will therefore have to be zero.
04

- Calculate the Dot Product

To find the dot product of the two vectors, multiply their individual components and add them. \( \vec{v} \cdot \vec{a} = (-g * v_{y0}+gtg) = 0 \). Solving this equation yields \( t = v_{y0}/g \). This means that when time \( t \) is equal to the initial vertical speed divided by the gravitational pull, the velocity and acceleration vectors are perpendicular.

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

How fast would a car have to round a 75 -m-radius turn for its acceleration to be numerically equal to that of gravity?

The table below lists position versus time for an object moving in the \(x-y\) plane, which is horizontal in this case. Make a plot of position \(y\) versus \(x\) to determine the nature of the object's path. Then determine the magnitudes of the object's velocity and acceleration. $$\begin{array}{ccc} \text { Time, } t(\mathrm{s}) & x(\mathrm{m}) & y(\mathrm{m}) \\ 0 & 0 & 0 \\ 0.10 & 0.65 & 0.09 \\ 0.20 & 1.25 & 0.33 \\ 0.30 & 1.77 & 0.73 \\ 0.40 & 2.17 & 1.25 \\ 0.50 & 2.41 & 1.85 \\ 0.60 & 2.50 & 2.50 \end{array}$$ $$\begin{array}{ccc} \text { Time, } t(\mathrm{s}) & x(\mathrm{m}) & y(\mathrm{m}) \\ 0.70 & 2.41 & 3.15 \\ 0.80 & 2.17 & 3.75 \\ 0.90 & 1.77 & 4.27 \\ 1.00 & 1.25 & 4.67 \\ 1.10 & 0.65 & 4.91 \\ 1.20 & 0.00 & 5.00 \end{array}$$

Can two vectors of equal magnitude sum to zero? How about two vectors of unequal magnitude?

You're in a bus moving with constant velocity on a level road when you throw a ball straight up. When the ball returns, does it land ahead of you, behind you, or back at your hand? Explain.

Derive a general formula for the horizontal distance covered by a projectile launched horizontally at speed \(v_{0}\) from height \(h\)
See all solutions

Recommended explanations on Physics Textbooks

Nuclear Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Radiation

Read Explanation

Physics of Motion

Read Explanation

Electricity and Magnetism

Read Explanation

Energy Physics

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.