Question

The position vector r(t) =<<n t, 1/ t^2, t^4>> describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t = sqrt3

Vectors and spaces
ANSWERED
asked 2021-03-08

The position vector \(\displaystyle{r}{\left({t}\right)}={\left\langle{\ln}\ {t},\frac{{1}}{{t}^{{2}}},{t}^{{4}}\right\rangle}\) describes the path of an object moving in space.
(a) Find the velocity vector, speed, and acceleration vector of the object.
(b) Evaluate the velocity vector and acceleration vector of the object at the given value of \(\displaystyle{t}=\sqrt{{3}}\)

Answers (1)

2021-03-09

a) Velocity vector is the derivative of position vector.
So to find velocity vector, differentiate position vector with respect to t. We get \(\displaystyle{v}{\left({t}\right)}={\left\langle\frac{{1}}{{5}},-\frac{{2}}{{t}^{{3}}},{4}{t}^{{3}}\right\rangle}\)
Now speed is the magnitude of velocity vector. That is,
Speed \(\displaystyle={\left|{\left|{v}\right|}\right|}\)
\(\displaystyle=\sqrt{{{\left(\frac{{1}}{{t}}\right)}^{{2}}+{\left(-\frac{{2}}{{t}^{{3}}}\right)}^{{2}}+{\left({4}{t}^{{3}}\right)}^{{2}}}}\)
\(\displaystyle=\sqrt{{\frac{{1}}{{t}^{{2}}}+\frac{{4}}{{t}^{{6}}}+{16}{t}^{{6}}}}\)
\(\displaystyle=\sqrt{{\frac{{{t}^{{4}}+{4}+{16}{t}^{{12}}}}{{t}^{{6}}}}}\)
\(\displaystyle=\frac{\sqrt{{{\left({16}{t}^{{12}}+{t}^{{4}}+{4}\right)}}}}{{t}^{{3}}}\)
Acceleration vector is the derivative of the velocity vector.
So to find acceleration vector, differentiate velocity vector with respect to t. We get,
\(a(t)=\left\langle-\frac{1}{t^{2}},\ -2(-\frac{3}{t^{4}}),\ 4(3t^{2}) \right\rangle\)

\(=\left\langle -\frac{1}{t^{2}},\ \frac{6}{t^{4}},\ 12t^{2}\right\rangle\)
b) Put \(\displaystyle{t}=\sqrt{{3}}\) in the velocity vector, we get
\(\displaystyle{v}{\left(\sqrt{{3}}\right)}={\left\langle\frac{{1}}{\sqrt{{3}}},-\frac{{2}}{{\left(\sqrt{{3}}\right)}^{{3}}},{4}{\left(\sqrt{{3}}\right)}^{{3}}\right\rangle}\)
\(\displaystyle={\left\langle\frac{{1}}{\sqrt{{3}}},-\frac{{2}}{{{3}\sqrt{{3}}}},{12}\sqrt{{3}}\right\rangle}\)
Put \(\displaystyle{t}=\sqrt{{3}}\) in the acceleration vector, we get
\(\displaystyle{a}{\left(\sqrt{{3}}\right)}={\left\langle-\frac{{1}}{{\left(\sqrt{{3}}\right)}^{{2}}},\frac{{6}}{{\left(\sqrt{{3}}\right)}^{{4}}},{12}{\left(\sqrt{{3}}\right)}^{{2}}\right\rangle}\)
\(\displaystyle={\left\langle-\frac{{1}}{{3}},\frac{{6}}{{9}},{12}{\left({3}\right)}\right\rangle}\)
\(\displaystyle={\left\langle-\frac{{1}}{{3}},\frac{{2}}{{3}},{36}\right\rangle}\)

0
 
Best answer

expert advice

Need a better answer?

Relevant Questions

asked 2021-01-28

Given the vector \(r(t) = { \cos T, \sin T, \ln ( \cos T) }\) and point (1, 0, 0) find vectors T, N and B at that point. \( Vector T is the unit tangent vector, so the derivative r(t) is needed. \( Vector N is the normal unit vector, and the equation for it uses the derivative of T(t). \( The B vector is the binormal vector, which is a crossproduct of T and N.

asked 2020-11-30

Consider \(\displaystyle{V}={s}{p}{a}{n}{\left\lbrace{\cos{{\left({x}\right)}}},{\sin{{\left({x}\right)}}}\right\rbrace}\) a subspace of the vector space of continuous functions and a linear transformation \(\displaystyle{T}:{V}\rightarrow{V}\) where \(\displaystyle{T}{\left({f}\right)}={f{{\left({0}\right)}}}\times{\cos{{\left({x}\right)}}}−{f{{\left(π{2}\right)}}}\times{\sin{{\left({x}\right)}}}.\) Find the matrix of T with respect to the basis \(\displaystyle{\left\lbrace{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}},{\cos{{\left({x}\right)}}}−{\sin{{\left({x}\right)}}}\right\rbrace}\) and determine if T is an isomorphism.

asked 2021-05-01

Identify the surface with the given vector equation. \(r(s,t)=\)\((s,t,t^2-s^2)\)
eliptic cylinder
circular paraboloid
hyperbolic paraboloid
plane
circular cylinder

asked 2021-01-15

It is given: \(\displaystyle∥{m}∥={4},∥{n}∥=\sqrt{{{2}}},⟨{m},{n}⟩={135}\) find the norm of the vector \(m+3n\).

asked 2021-02-25

Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u,w) where \(u \in U\) and \(w \in W\). Show that V is a vector space over K with addition in V and scalar multiplication on V defined by
\((u,w)+(u',w')=(u+u',w+w')\ and\ k(u,w)=(ku,kw)\)
(This space V is called the external direct product of U and W.)

asked 2020-12-29
The position vector \(\displaystyle{r}{\left({t}\right)}={\left\langle{\ln{{t}}},\frac{{1}}{{t}^{{2}}},{t}^{{4}}\right\rangle}\) describes the path of an object moving in space.
(a) Find the velocity vector, speed, and acceleration vector of the object.
(b) Evaluate the velocity vector and acceleration vector of the object at the given value of \(\displaystyle{t}=\sqrt{{3}}\)
asked 2021-05-27

Vector Cross Product
Let vectors \(A=(1,0,-3), B =(-2,5,1),\ and\ C =(3,1,1)\). Calculate the following, expressing your answers as ordered triples (three comma-separated numbers).
(C) \((2\bar B)(3\bar C)\)
(D) \((\bar B)(\bar C)\)
(E) \(\overrightarrow A(\overrightarrow B \times \overrightarrow C)\)
(F)If \(\bar v_1 \text{ and } \bar v_2\) are perpendicular, \(|\bar v_1 \times \bar v_2|\)
(G) If \(\bar v_1 \text{ and } \bar v_2\) are parallel, \(|\bar v_1 \times \bar v_2|\)

asked 2021-03-02

Let u,\(v_1\) and \(v_2\) be vectors in \(R^3\), and let \(c_1\) and \(c_2\) be scalars. If u is orthogonal to both \(v_1\) and \(v_2\), prove that u is orthogonal to the vector \(c_1v_1+c_2v_2\).

asked 2021-01-02

Write the vector v in the form ai + bj, given its magnitude ∥v∥∥v∥ and the angle \(\alpha\) it makes with the positive x-axis. \(∥v∥=8,\alpha=45*∥v∥=8,\alpha=45\)

asked 2021-01-31

Let \(\displaystyle{B}={\left\lbrace{v}{1},{v}{2},\ldots,{v}{m}\right\rbrace}\) be a basis for \(R^{m}\). Suppose kvm is a linear combination of \(v1, v2, \cdots, vm-1\) for some scalar k. What can be said about the possible value(s) of k?

You might be interested in

...