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Chapter 12: Problem 13

Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $$\mathbf{r}(t)=\left\langle t^{2}+3, t^{2}+10, \frac{1}{2} t^{2}\right\rangle, \text { for } t \geq 0$$

Short Answer

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Question: Given the position function for an object, \(\mathbf{r}(t)=\left\langle t^{2}+3, t^{2}+10, \frac{1}{2}t^{2}\right\rangle\), find: a. The velocity and speed of the object. b. The acceleration of the object. Answer: a. Velocity: \(\mathbf{v}(t)=\left\langle 2t, 2t, t\right\rangle\), speed: \(| \mathbf{v}(t) |=3t \text{ for } t \geq 0\) b. Acceleration: \(\mathbf{a}(t)=\left\langle 2, 2, 1\right\rangle\)

Step by step solution

01

Calculate the velocity vector

First, we need to find the velocity vector, which is the derivative of the position vector with respect to time. The position vector is given by \(\mathbf{r}(t)=\left\langle t^{2}+3, t^{2}+10, \frac{1}{2}t^{2}\right\rangle\). Taking the derivative of each component with respect to time gives us the velocity vector: $$\mathbf{v}(t)=\frac{d\mathbf{r}}{dt}=\left\langle \frac{d(t^2 + 3)}{dt}, \frac{d(t^2 + 10)}{dt}, \frac{d(\frac{1}{2}t^2)}{dt} \right\rangle$$ $$\mathbf{v}(t)=\left\langle 2t, 2t, t\right\rangle$$
02

Calculate the speed

Next, we need to find the speed, which is the magnitude of the velocity vector. The magnitude of a vector \(\mathbf{v}=\left\langle a, b, c \right\rangle\) is given by \(|\mathbf{v}|=\sqrt{a^2+b^2+c^2}\). So, the speed of the object is: $$|\mathbf{v}|=\sqrt{(2t)^2+(2t)^2+(t)^2}=\sqrt{4t^2+4t^2+t^2}=\sqrt{9t^2}=3t \text{ for } t \geq 0$$
03

Calculate the acceleration vector

Now, we need to find the acceleration vector, which is the derivative of the velocity vector with respect to time. The velocity vector is \(\mathbf{v}(t)=\left\langle 2t, 2t, t\right\rangle\). Taking the derivative of each component with respect to time gives us the acceleration vector: $$\mathbf{a}(t)=\frac{d\mathbf{v}}{dt}=\left\langle \frac{d(2t)}{dt}, \frac{d(2t)}{dt}, \frac{d(t)}{dt} \right\rangle$$ $$\mathbf{a}(t)=\left\langle 2, 2, 1\right\rangle$$ So, the velocity and speed of the object are given by: a. Velocity: \(\mathbf{v}(t)=\left\langle 2t, 2t, t\right\rangle\), speed: \(| \mathbf{v}(t) |=3t \text{ for } t \geq 0\) b. Acceleration: \(\mathbf{a}(t)=\left\langle 2, 2, 1\right\rangle\)

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Most popular questions from this chapter

Assume that \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in \(\mathrm{R}^{3}\) that form the sides of a triangle (see figure). Use the following steps to prove that the medians intersect at a point that divides each median in a 2: 1 ratio. The proof does not use a coordinate system. a. Show that \(\mathbf{u}+\mathbf{v}+\mathbf{w}=\mathbf{0}\) b. Let \(\mathbf{M}_{1}\) be the median vector from the midpoint of \(\mathbf{u}\) to the opposite vertex. Define \(\mathbf{M}_{2}\) and \(\mathbf{M}_{3}\) similarly. Using the geometry of vector addition show that \(\mathbf{M}_{1}=\mathbf{u} / 2+\mathbf{v} .\) Find analogous expressions for \(\mathbf{M}_{2}\) and \(\mathbf{M}_{3}\) c. Let \(a, b,\) and \(c\) be the vectors from \(O\) to the points one-third of the way along \(\mathbf{M}_{1}, \mathbf{M}_{2},\) and \(\mathbf{M}_{3},\) respectively. Show that \(\mathbf{a}=\mathbf{b}=\mathbf{c}=(\mathbf{u}-\mathbf{w}) / 3\) d. Conclude that the medians intersect at a point that divides each median in a 2: 1 ratio.

Maximum curvature Consider the "superparabolas" \(f_{n}(x)=x^{2 n},\) where \(n\) is a positive integer. a. Find the curvature function of \(f_{n},\) for \(n=1,2,\) and 3 b. Plot \(f_{n}\) and their curvature functions, for \(n=1,2,\) and 3 and check for consistency. c. At what points does the maximum curvature occur, for \(n=1,2,3 ?\) d. Let the maximum curvature for \(f_{n}\) occur at \(x=\pm z_{n} .\) Using either analytical methods or a calculator determine \(\lim _{n \rightarrow \infty} z_{n}\) Interpret your result.

Alternative derivation of the curvature Derive the computational formula for curvature using the following steps. a. Use the tangential and normal components of the acceleration to show that \(\left.\mathbf{v} \times \mathbf{a}=\kappa|\mathbf{v}|^{3} \mathbf{B} . \text { (Note that } \mathbf{T} \times \mathbf{T}=\mathbf{0} .\right)\) b. Solve the equation in part (a) for \(\kappa\) and conclude that \(\kappa=\frac{|\mathbf{v} \times \mathbf{a}|}{\left|\mathbf{v}^{3}\right|},\) as shown in the text.

Find the points (if they exist) at which the following planes and curves intersect. $$y=1 ; \mathbf{r}(t)=\langle 10 \cos t, 2 \sin t, 1\rangle, \text { for } 0 \leq t \leq 2 \pi$$

An object moves along a path given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) for \(0 \leq t \leq 2 \pi\) a. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the path is a circle (in a plane)? b. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the path is an ellipse (in a plane)?
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