/*! 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 65 Speed on an ellipse An object mo... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 65

Speed on an ellipse An object moves along an ellipse given by the function \(\mathbf{r}(t)=\langle a \cos t, b \sin t\rangle,\) for \(0 \leq t \leq 2 \pi,\) where \(a>0\) and \(b>0\). a. Find the velocity and speed of the object in terms of \(a\) and \(b\), for \(0 \leq t \leq 2 \pi\). b. With \(a=1\) and \(b=6,\) graph the speed function, for \(0 \leq t \leq 2 \pi .\) Mark the points on the trajectory at which the speed is a minimum and a maximum. c. Is it true that the object speeds up along the flattest (straightest) parts of the trajectory and slows down where the curves are sharpest? d. For general \(a\) and \(b\), find the ratio of the maximum speed to the minimum speed on the ellipse (in terms of \(a\) and \(b\) ).

Short Answer

Expert verified
#tag_title# b. Graph the speed function and mark the points #tag_content# To graph the speed function, let's first simplify it: $$ \text{speed} = \sqrt{(-a\sin{t})^2 + (b\cos{t})^2} = \sqrt{a^2\sin^2{t} + b^2\cos^2{t}}. $$ Now, we need to find the minimum and maximum speed. Since the speed function is squared inside the square root, we can analyze the function inside the square root to determine the maximum and minimum values. Let \(g(t) = a^2\sin^2{t} + b^2\cos^2{t}\). We want to find the maximum and minimum values of \(g(t)\), which correspond to the maximum and minimum speeds. We can use the first and second derivative tests to find these values. First, let's find the critical points of \(g(t)\). We can do this by taking the first derivative of \(g(t)\) and setting it equal to zero. $$ g'(t) = 2a^2\sin{t}\cos{t} - 2b^2\sin{t}\cos{t} = 0, $$ which yields, $$ \sin{t}\cos{t} \left( a^2 - b^2 \right) = 0. $$ Thus, the critical points are at \(t = k\pi / 2\), where \(k\) is an integer. Now, let's use the second derivative test to determine if these critical points correspond to minima or maxima: $$ g''(t) = 2a^2\left( \cos^2{t} - \sin^2{t} \right) - 2b^2\left( \cos^2{t} - \sin^2{t} \right). $$ At \(t = 0\), we have \(g''(t) = 2a^2 - 2b^2 > 0\) if \(a > b\) and \(g''(t) < 0\) if \(a < b\). Thus, if \(a > b\), there is a minimum speed at \(t = 0\), and if \(a < b\), there is a maximum speed at \(t = 0\). Similarly, at \(t = \pi / 2\), we have \(g''(t) = -2a^2 + 2b^2 < 0\) if \(a > b\) and \(g''(t) > 0\) if \(a < b\). So, if \(a > b\), there is a maximum speed at \(t = \pi / 2\), and if \(a < b\), there is a minimum speed at \(t = \pi / 2\). We can mark these points on the ellipse's trajectory to illustrate the minimum and maximum speeds.

Step by step solution

01

a. Find the velocity and speed

To find the velocity, we need to take the derivative of the position function. The position function is given by \(\mathbf{r}(t) = \langle a\cos t, b\sin t \rangle\). Taking the derivative component-wise, $$ \mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} = \left\langle -a\sin t, b\cos t \right\rangle. $$ Next, we'll find the speed by finding the magnitude of the velocity vector, $$ \text{speed} = \lVert\mathbf{v}(t)\rVert = \sqrt{(-a\sin{t})^2 + (b\cos{t})^2}. $$

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Key Concepts

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

Parametric Equations
Parametric equations express a set of related quantities as explicit functions of an independent variable, known as a parameter. In the case of motion on an ellipse, parametric equations provide a convenient way to describe the position of an object at any given time. For example, consider the ellipse defined by the function \(\mathbf{r}(t)=\langle a \cos t, b \sin t\rangle\), where \(a\) and \(b\) are positive constants and \(t\) represents time. This pair of equations explicitly gives the \(x\) and \(y\) coordinates of an object in terms of the cosine and sine functions, respectively. The parameter \(t\) typically varies within a certain range, which in this case is from \(0\) to \(2\pi\).

Understanding these parametric equations is essential because they form the basis for analyzing the object's movement, such as calculating its velocity and acceleration at any point in time. Plus, they're incredibly versatile, allowing for the description of even the most complex curves and motions quite succinctly.
Derivative of Position Function
The derivative of a position function represents the object's velocity. It gives us the rate at which the position is changing with respect to time. Taking the derivative of the parametric equations of the ellipse \(\mathbf{r}(t)=\langle a \cos t, b \sin t\rangle\), we calculate the components of the velocity vector. Performing the differentiation component-wise, we obtain \(\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} = \left\langle -a\sin t, b\cos t \right\rangle\).

This step is crucial for understanding how fast the object is moving and in what direction at any given moment. The velocity vector not only tells us the speed of the object but also the direction of movement, which is particularly useful when analyzing motion along curves or irregular paths, such as an ellipse.
Velocity Vector
A velocity vector is a vector that describes both the speed and the direction of an object's motion. It is obtained by differentiating the position function with respect to time. For our elliptical motion, the velocity vector is \(\mathbf{v}(t) = \left\langle -a\sin t, b\cos t \right\rangle\), derived from the position function \(\mathbf{r}(t)\). Each component of this vector represents the rate of change of the object鈥檚 position in the respective \(x\) and \(y\) directions.

The direction of the velocity vector at any point in time is tangential to the path of the object. In other words, if you were to draw a line that just touches the curve at a point without crossing it, this line would be in the direction of the velocity vector. So, understanding velocity vectors is essential for predicting the future position of the object, as it quite literally points to where the object is heading next.
Magnitude of Velocity
The magnitude of velocity, commonly referred to as speed, is a scalar quantity that measures how fast an object is moving regardless of its direction. Mathematically, it is found by taking the norm (or magnitude) of the velocity vector. For our object moving along an ellipse, the speed would be calculated using the formula \(\text{speed} = \lVert\mathbf{v}(t)\rVert = \sqrt{(-a\sin{t})^2 + (b\cos{t})^2}\).

This equation simplifies to \(\text{speed} = \sqrt{a^2\sin^2{t} + b^2\cos^2{t}}\), which gives the object's speed at any time \(t\). It's important to note that, even though the path is elliptical, the speed is not constant because the ellipse does not have a uniform curvature. The speed varies depending on the position of the object on the ellipse, being different at different points. Such analysis of speed allows one to understand the dynamics of the object's motion comprehensively and predict its kinetic energy.

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Most popular questions from this chapter

Designing a baseball pitch A baseball leaves the hand of a pitcher 6 vertical feet above and 60 horizontal feet from home plate. Assume the coordinate axes are oriented as shown in the figure. Figure cannot copy a. Suppose a pitch is thrown with an initial velocity of (130,0,-3) ft/s (about \(90 \mathrm{mi} / \mathrm{hr}\) ). In the absence of all forces except gravity, how far above the ground is the ball when it crosses home plate and how long does it take the pitch to arrive? b. What vertical velocity component should the pitcher use so that the pitch crosses home plate exactly 3 ft above the ground? c. A simple model to describe the curve of a baseball assumes the spin of the ball produces a constant sideways acceleration (in the \(y\) -direction) of \(c \mathrm{ft} / \mathrm{s}^{2} .\) Suppose a pitcher throws a curve ball with \(c=8 \mathrm{ft} / \mathrm{s}^{2}\) (one fourth the acceleration of gravity). How far does the ball move in the \(y\) -direction by the time it reaches home plate, assuming an initial velocity of \((130,0,-3) \mathrm{ft} / \mathrm{s} ?\) d. In part (c), does the ball curve more in the first half of its trip to the plate or in the second half? How does this fact affect the batter? e. Suppose the pitcher releases the ball from an initial position of \langle 0,-3,6\rangle with initial velocity \(\langle 130,0,-3\rangle .\) What value of the spin parameter \(c\) is needed to put the ball over home plate passing through the point (60,0,3)\(?\)

Trajectory properties Find the time of flight, range, and maximum height of the following two-dimensional trajectories, assuming no forces other than gravity. In each case, the initial position is (0,0) and the initial velocity is \(\mathbf{v}_{0}=\left\langle u_{0}, v_{0}\right\rangle\). $$\left\langle u_{0}, v_{0}\right\rangle=\langle 40,80\rangle \mathrm{m} / \mathrm{s}$$

Suppose the vector-valued function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) is smooth on an interval containing the point \(t_{0}\) The line tangent to \(\mathbf{r}(t)\) at \(t=t_{0}\) is the line parallel to the tangent vector \(\mathbf{r}^{\prime}\left(t_{0}\right)\) that passes through \(\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right) .\) For each of the following functions, find an equation of the line tangent to the curve at \(t=t_{0} .\) Choose an orientation for the line that is the same as the direction of \(\mathbf{r}^{\prime}.\) $$\mathbf{r}(t)=\left\langle 3 t-1,7 t+2, t^{2}\right\rangle ; t_{0}=1$$

Nonuniform straight-line motion Consider the motion of an object given by the position function $$\mathbf{r}(t)=f(t)\langle a, b, c\rangle+\left\langle x_{0}, y_{0}, z_{0}\right\rangle, \quad \text { for } t \geq 0$$ where \(a, b, c, x_{0}, y_{0},\) and \(z_{0}\) are constants, and \(f\) is a differentiable scalar function, for \(t \geq 0\) a. Explain why \(r\) describes motion along a line. b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?

Find the point P on the curve \(\mathbf{r}(t)\) that lies closest to \(P_{0}\) and state the distance between \(P_{0}\) and \(P\). $$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k} ; P_{0}(1,1,3)$$
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