3.4 Motion in space  (Page 9/17)

Key concepts

• If $\text{r}\left(t\right)$ represents the position of an object at time t , then $\text{r}\prime \left(t\right)$ represents the velocity and $\text{r″}\left(t\right)$ represents the acceleration of the object at time t. The magnitude of the velocity vector is speed.
• The acceleration vector always points toward the concave side of the curve defined by $\text{r}\left(t\right).$ The tangential and normal components of acceleration $a_{\text{T}}$ and $a_{\text{N}}$ are the projections of the acceleration vector onto the unit tangent and unit normal vectors to the curve.
• Kepler’s three laws of planetary motion describe the motion of objects in orbit around the Sun. His third law can be modified to describe motion of objects in orbit around other celestial objects as well.
• Newton was able to use his law of universal gravitation in conjunction with his second law of motion and calculus to prove Kepler’s three laws.

Key equations

• Velocity
  $\text{v}\left(t\right)=r^{\prime }\left(t\right)$
• Acceleration
  $\text{a}\left(t\right)=v^{\prime }\left(t\right)=\text{r″}\left(t\right)$
• Speed
  $v\left(t\right)=\Vert \text{v}\left(t\right)\Vert =\Vert r^{\prime }\left(t\right)\Vert =\frac{ds}{dt}$
• Tangential component of acceleration
  $a_{\text{T}}=\text{a}·\text{T}=\frac{\text{v}·\text{a}}{\Vert \text{v}\Vert }$
• Normal component of acceleration
  $a_{\text{N}}=\text{a}·\text{N}=\frac{\Vert \text{v}\times \text{a}\Vert }{\Vert \text{v}\Vert }=\sqrt{{\Vert \text{a}\Vert }^2-a_{}^{\text{T}}}$

Given $\text{r}(t)=(3t^2-2)\text{i}+(2t-\text{sin}(t))\text{j},$ find the velocity of a particle moving along this curve.

$\text{v}(t)=(6t)\text{i}+(2-\text{cos}(t))\text{j}$

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Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter t .

$\text{r}(t)=2\text{cos}t\text{j}+3\text{sin}t\text{k}.$ The graph is shown here:

$v(t)=\mathrm{-2}\sin tj+3\cos tk,$ $\text{a}(t)=\mathrm{-2}\text{cos}t\text{j}-3\text{sin}t\text{k},$ $\text{speed}=\sqrt{4{\text{sin}}^2(t)+9{}^{\text{cos}}(t)}$

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Find the velocity, acceleration, and speed of a particle with the given position function.

$\text{r}(t)=⟨\text{sin}t,t,\text{cos}t⟩.$ The graph is shown here:

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Consider the motion of a point on the circumference of a rolling circle. As the circle rolls, it generates the cycloid $\text{r}(t)=\left(\omega t-\text{sin}(\omega t)\right)\text{i}+\left(1-\text{cos}(\omega t)\right)\text{j},$ where $\omega$ is the angular velocity of the circle and b is the radius of the circle:

A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector $\text{r}(t)=(3\text{cos}t)\text{i}+(3\text{sin}t)\text{j}+t^2\text{k}.$ The path is similar to that of a helix, although it is not a helix. The graph is shown here:

Find the following quantities:

Given that $\text{r}(t)=⟨e^{\mathrm{-5}t}\text{sin}t,e^{\mathrm{-5}t}\text{cos}t,4e^{\mathrm{-5}t}⟩$ is the position vector of a moving particle, find the following quantities: