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Chapter 12: Problem 15

Consider the following position functions. a. Find the velocity and speed of the object. b. Find the acceleration of the object. $$\mathbf{r}(t)=\langle 3+t, 2-4 t, 1+6 t\rangle, \text { for } t \geq 0$$

Short Answer

Expert verified
Answer: The velocity function is $$\mathbf{v}(t) = \langle 1, -4, 6\rangle$$, the speed is $$\sqrt{53}$$, and the acceleration function is $$\mathbf{a}(t) = \langle0, 0, 0\rangle$$.

Step by step solution

01

Differentiate the position function to find the velocity function

Recall that the velocity function is the first derivative of the position function with respect to time (t). So we need to differentiate each component of the position vector with respect to t: $$\mathbf{r}(t)=\langle 3+t, 2-4 t, 1+6 t\rangle$$ $$\frac{d\mathbf{r}}{dt} = \left\langle \frac{d}{dt}(3+t), \frac{d}{dt}(2-4t), \frac{d}{dt}(1+6t) \right\rangle$$ $$\mathbf{v}(t) = \left\langle 1, -4, 6 \right\rangle$$ Therefore, the velocity function is: $$\mathbf{v}(t) = \langle 1, -4, 6\rangle$$
02

Find the speed of the object

The speed of the object is given by the magnitude of the velocity vector. Calculate the magnitude as follows: $$\text{speed} = \|\mathbf{v}(t)\| = \sqrt{(1)^2 + (-4)^2 + (6)^2} = \sqrt{1 + 16 + 36} = \sqrt{53}$$ Therefore, the speed of the object is: $$\text{speed} = \sqrt{53}$$
03

Differentiate the velocity function to find the acceleration function

Recall that the acceleration function is the first derivative of the velocity function with respect to time (t). So we need to differentiate each component of the velocity vector with respect to t: $$\mathbf{v}(t) = \langle 1, -4, 6\rangle$$ $$\frac{d\mathbf{v}}{dt} = \left\langle \frac{d}{dt}(1), \frac{d}{dt}(-4), \frac{d}{dt}(6) \right\rangle$$ $$\mathbf{a}(t) = \left\langle 0, 0, 0 \right\rangle$$ Therefore, the acceleration function is: $$\mathbf{a}(t) = \langle0, 0, 0\rangle$$ To summarize: a. The velocity of the object is given by the function $$\mathbf{v}(t) = \langle 1, -4, 6\rangle$$, and the speed of the object is $$\sqrt{53}$$. b. The acceleration of the object is given by the function $$\mathbf{a}(t) = \langle0, 0, 0\rangle$$.

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Most popular questions from this chapter

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.

Let \(\mathbf{v}=\langle a, b, c\rangle\) and let \(\alpha, \beta\) and \(\gamma\) be the angles between \(\mathbf{v}\) and the positive \(x\) -axis, the positive \(y\) -axis, and the positive \(z\) -axis, respectively (see figure). a. Prove that \(\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1\) b. Find a vector that makes a \(45^{\circ}\) angle with i and \(\mathbf{j}\). What angle does it make with k? c. Find a vector that makes a \(60^{\circ}\) angle with i and \(\mathbf{j}\). What angle does it make with k? d. Is there a vector that makes a \(30^{\circ}\) angle with i and \(\mathbf{j}\) ? Explain. e. Find a vector \(\mathbf{v}\) such that \(\alpha=\beta=\gamma .\) What is the angle?

A pair of lines in \(\mathbb{R}^{3}\) are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect. determine the point(s) of intersection. $$\begin{aligned} &\mathbf{r}(t)=\langle 1+2 t, 7-3 t, 6+t\rangle;\\\ &\mathbf{R}(s)=\langle-9+6 s, 22-9 s, 1+3 s\rangle \end{aligned}$$

\(\mathbb{R}^{2}\) Consider the vectors \(\mathbf{I}=\langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle\) and \(\mathbf{J}=\langle-1 / \sqrt{2}, 1 / \sqrt{2}\rangle\). Write the vector \langle 2,-6\rangle in terms of \(\mathbf{I}\) and \(\mathbf{J}\).

A pair of lines in \(\mathbb{R}^{3}\) are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect. determine the point(s) of intersection. $$\begin{aligned} &\mathbf{r}(t)=\langle 3+4 t, 1-6 t, 4 t\rangle;\\\ &\mathbf{R}(s)=\langle-2 s, 5+3 s, 4-2 s\rangle \end{aligned}$$
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