/*! 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 46 A projectile has been launched f... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 46

A projectile has been launched from the ground with initial velocity 88 ft per sec. The parametric equations modeling the path of the projectile are supplied. (a) Graph the parametric equations. (b) Approximate \(\theta\), the angle the projectile makes with the horizontal at launch, to the nearest tenth of a degree. (c) Based on your answer to part (b), write parametric equations for the projectile using the cosine and sine functions. $$x=82.69295063 t, y=-16 t^{2}+30.09777261 t$$

Short Answer

Expert verified
Graph the parametric equations, calculate the angle using \(\theta = \text{tan}^{-1}\left(\frac{30.09777261}{82.69295063}\right)\), and write the parametric equations with the angle and initial velocity.

Step by step solution

01

Graph the Parametric Equations

Use a graphing calculator or software to plot the given parametric equations. Input the equations: \(x=82.69295063 t\) and \(y=-16 t^{2}+30.09777261 t\). Observe the resulting trajectory of the projectile.
02

Identify Initial Conditions

Identify the initial horizontal velocity \(v_{x}\) and the initial vertical velocity \(v_{y}\) from the given parametric equations. The initial velocities can be found from the coefficients of \(t\) in both equations. Here, \(v_{x} = 82.69295063\) ft/sec and \(v_{y} = 30.09777261\) ft/sec.
03

Calculate Launch Angle \(\theta\)

Use the formula for the launch angle: \[ \theta = \text{tan}^{-1} \left(\frac{v_{y}}{v_{x}}\right) \] Substituting the values, we get: \[ \theta = \text{tan}^{-1} \left(\frac{30.09777261}{82.69295063}\right) \] Calculate this using a scientific calculator to find \(\theta\) to the nearest tenth of a degree.
04

Write Parametric Equations Using Cosine and Sine Functions

Based on the calculated value of \(\theta\), write the parametric equations in terms of the initial velocity and the angle components. The general form is: \[ x(t) = v_{0} \cos (\theta) t \]\[ y(t) = v_{0} \sin (\theta) t - \frac{1}{2} g t^{2} \] Use the initial velocity \(v_{0} = 88\) ft/sec and the angle \(\theta\) found in the previous step, and \(g = 32\) ft/sec虏.

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!

Key Concepts

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

Parametric Equations
Parametric equations offer a way to describe the position of a projectile in motion by defining both its x and y coordinates as functions of time, denoted as t. This approach effectively breaks down the projectile's path into two simpler, related equations.
For the given problem, we have:
  • x(t) = 82.69295063 t
  • y(t) = -16 t虏 + 30.09777261 t
Graphing these parametric equations helps visualize the projectile鈥檚 trajectory by plotting the x and y coordinates simultaneously as t varies.
Launch Angle Calculation
The launch angle \(\theta \) is crucial in determining the initial direction of a projectile's motion. It can be calculated using the ratio of the initial vertical velocity \(v_y\) to the initial horizontal velocity \(v_x\).
Using the formula:
\[ \theta = \tan^{-1} \left( \frac{v_y}{v_x} \right) \]
We substitute our values \(v_y = 30.09777261\) and \(v_x = 82.69295063\):
\[ \theta = \tan^{-1} \left( \frac{30.09777261}{82.69295063} \right) \]
This equals approximately 20 degrees when you compute it using a calculator, rounding to the nearest tenth of a degree.
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are pivotal in breaking down the initial velocity into horizontal and vertical components. These components are derived from the initial velocity \(v_0\) and the launch angle \(\theta\).
The horizontal component \(v_x\) is calculated using the cosine function:
\[ v_x = v_0 \cos(\theta) \]
The vertical component \(v_y\) uses the sine function:
\[ v_y = v_0 \sin(\theta) \]
By splitting the velocity this way, we can use these components in parametric equations to model the projectile鈥檚 motion accurately.
Initial Velocity Components
The initial velocity of the projectile, given as 88 ft/sec, can be dissected into two useful components:
\[ v_x = 88 \cos(\theta) \] \[ v_y = 88 \sin(\theta) \]
Given our calculated angle of approximately 20 degrees, we find:
  • Horizontal component: \(\text{cos}(20^\text{掳}) \times 88 \approx 82.7\) ft/sec
  • Vertical component: \(\text{sin}(20^\text{掳}) \times 88 \approx 30.1\) ft/sec
These are practical and significant for understanding how the projectile will move over time.
Graphing Trajectories
Visualizing the projectile's motion through a graph helps in comprehending its path. To graph the equations:
  • Input \(x(t) = 82.69295063 t\)
  • Input \(y(t) = -16 t^2 + 30.09777261 t\)
By plotting these equations, we observe the characteristic curved trajectory, showcasing how the object rises to a peak and then descends back to the ground. This visual representation confirms our calculations regarding velocities and angle, and helps in appreciating the underlying physics of projectile motion.

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

Solve each problem. The polar equation $$r=\frac{a\left(1-e^{2}\right)}{1+e \cos \theta}$$ can be used to graph the orbits of the satellites of our sun, where \(a\) is the average distance in astronomical units from the sun and \(e\) is a constant called the eccentricity. The sun will be located at the pole. The table lists the values of \(a\) and \(e\). (a) Graph the orbits of the four closest satellites on the same polar grid. Choose a viewing window that results in a graph with nearly circular orbits. (b) Plot the orbits of Earth, Jupiter, Uranus, and Pluto on the same polar grid. How does Earth's distance from the sun compare to the others' distances from the sun? (c) Use graphing to determine whether or not Pluto is always farthest from the sun. $$\begin{array}{|c|c|c|}\hline \text { Satellite } & a & e \\\\\hline \text { Mercury } & 0.39 & 0.206 \\\\\hline \text { Venus } & 0.78 & 0.007 \\\\\hline \text { Earth } & 1.00 & 0.017 \\\\\hline \text { Mars } & 1.52 & 0.093 \\\\\hline \text { Jupiter } & 5.20 & 0.048 \\\\\hline \text { Saturn } & 9.54 & 0.056 \\\\\hline \text { Uranus } & 19.20 & 0.047 \\\\\hline \text { Neptune } & 30.10 & 0.009 \\\\\hline \text { Pluto } & 39.40 & 0.249 \\\\\hline\end{array}$$

Concept Check The complex number \(z,\) where \(z=x+y i,\) can be graphed in the plane as \((x, y) .\) Describe the graphs of all complex numbers z satisfying the conditions. The imaginary part of \(z\) is 1

For each rectangular equation, give its equivalent polar equation and sketch its graph. $$x^{2}+y^{2}=9$$

For each pair of polar coordinates, ( \(a\) ) plot the point, ( \(b\) ) give two other pairs of polar coordinates for the point, and ( \(c\) ) give the rectangular coordinates for the point. $$\left(3, \frac{5 \pi}{3}\right)$$

Give a complete graph of each polar equation. Also identify the type of polar graph. $$r^{2}=4 \sin 2 \theta$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

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.