3.4 Motion in space  (Page 11/17)

Find the time in years it takes the dwarf planet Pluto to make one orbit about the Sun given that $a=39.5$ A.U.

Show that the particle moves on a circular cone.

Find the angle between the velocity and acceleration vectors when $t=1.5.$

Find the tangential and normal components of acceleration when $t=1.5.$

A parametric equation that passes through points P and Q can be given by $\text{r}(t)=⟨t^2,3t+1,t-2⟩,$ where $P(1,4,\mathrm{-1})$ and $Q(16,11,2).$

$\frac{d}{dt}\left[\text{u}(t)\times \text{u}(t)\right]=2{\text{u}}^{\prime }(t)\times \text{u}(t)$

The curvature of a circle of radius $r$ is constant everywhere. Furthermore, the curvature is equal to $1\text{/}r.$

The speed of a particle with a position function $\text{r}(t)$ is $\left({\text{r}}^{\prime }(t)\right)\text{/}\left(\left|{\text{r}}^{\prime }(t)\right|\right).$

$\text{r}(t)=⟨\text{sin}(t),\text{ln}(t),\sqrt{t}⟩$

$\text{r}(t)=⟨e^t,\frac{1}{\sqrt{4-t}},\text{sec}(t)⟩$

[T] $\text{r}(t)=⟨t^2,t^3⟩$

[T] $\text{r}(t)=⟨\text{sin}\left(20t\right)e^{\text{−}t},\text{cos}\left(20t\right)e^{\text{−}t},e^{\text{−}t}⟩$

Intersection of the cylinder $x^2+y^2=4$ with the plane $x+z=6$

Intersection of the cone $z=\sqrt{x^2+y^2}$ and plane $z=y-4$

$\text{u}(t)=⟨e^t,e^{\text{−}t}⟩$

$\text{u}(t)=⟨t^2,2t+6,4t^5-12⟩$

${\displaystyle \int \left(\text{tan}(t)\text{sec}(t)\text{i}-t{e}^{3t}\text{j}\right)dt}$

${\displaystyle \underset{1}{\overset{4}{\int }}\text{u}(t)dt},$ with $\text{u}(t)=⟨\frac{\text{ln}(t)}{t},\frac{1}{\sqrt{t}},\text{sin}\left(\frac{t\pi }{4}\right)⟩$

$\text{r}(t)=⟨3(t),4\text{cos}(t),4\text{sin}\left(t\right)⟩$ for $1\le t\le 4$

$\text{r}(t)=2\text{i}+\text{t}\text{j}+3t^2\text{k}$ for $0\le t\le 1$

$\text{r}(t)=2t\text{i}+(4t-5)\text{j}+(1-3t)\text{k}$

$\text{r}(t)=\text{cos}(2t)\text{i}+8t\text{j}-\text{sin}(2t)\text{k}$

$\text{r}(t)=(2\text{sin}t)\text{i}-4t\text{j}+(2\text{cos}t)\text{k}$

$\text{r}(t)=\sqrt{2}{e}^t\text{i}+\sqrt{2}{e}^{\text{−}t}\text{j}+2t\text{k}$

Find the unit tangent vector, the unit normal vector, and the binormal vector for $\text{r}(t)=2\text{cos}t\text{i}+3t\text{j}+2\text{sin}t\text{k}.$

Find the tangential and normal acceleration components with the position vector $\text{r}(t)=⟨\text{cos}t,\text{sin}t,e^t⟩.$

A Ferris wheel car is moving at a constant speed $v$ and has a constant radius $r.$ Find the tangential and normal acceleration of the Ferris wheel car.

The position of a particle is given by $\text{r}(t)=⟨t^2,\text{ln}\left(t\right),\text{sin}\left(\pi t\right)⟩,$ where $t$ is measured in seconds and $\text{r}$ is measured in meters. Find the velocity, acceleration, and speed functions. What are the position, velocity, speed, and acceleration of the particle at 1 sec?

Find the velocity vector function $\text{v}(t).$

Find the position vector $\text{r}(t)$ and the parametric representation for the position.

At what angle do you need to fire the cannonball for the horizontal distance to be greatest? What is the total distance it would travel?