Speed and arc length For the following trajectories, find the speed associated with the trajectory, and then find the length of the trajectory on the given interval. r(t) = ⟨2t3, -t3, 5t3⟩, for 0 ≤ t ≤ 4

Speed and arc length For the following trajectories, find the speed associated with the trajectory, and then find the length of the trajectory on the given interval. r(t) = ⟨2t3, -t3, 5t3⟩, for 0 ≤ t ≤ 4

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 90E

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Speed and arc length For the following trajectories, find the speed associated with the trajectory, and then find the length of the trajectory on the given interval.

r(t) = ⟨2t³, -t³, 5t³⟩, for 0 ≤ t ≤ 4

Expert Solution

Step 1

Find the speed associated with the trajectory, and then find the length of the trajectory on the given interval.

Step 2

The speed of the curve r(t) is $|v (t)| = |r' (t)|$

Compute the value of $r' (t)$

$\begin{array}{rcl} r' (t) & = & <\frac{d}{d t} (2 t^{3}), \frac{d}{d t} (- t^{3}), \frac{d}{d t} (5 t^{3})> \\ = & <6 t^{2}, - 3 t^{2}, 15 t^{2}> \end{array}$

Thus, the components of $r' (t)$ is $\begin{array}{rcl} ⟨6 t^{2}, - 3 t^{2}, 15 t^{2}⟩ \end{array}$ .

Use speed formula to compute the speed associated with the trajectory.

$\begin{array}{rcl} |v (t)| & = & |<6 t^{2}, - 3 t^{2}, 15 t^{2}>| \\ = & \sqrt{{(6 t^{2})}^{2} + (- 3 t^{2}) + {(15 t^{2})}^{2}} \\ = & \sqrt{270 t^{4}} \\ = & 3 \sqrt{30} t^{2} \end{array}$

Thus, the speed associated with the trajectory is $\begin{array}{rcl} 3 \end{array} \begin{array}{rcl} \sqrt{30} \end{array} \begin{array}{rcl} t \end{array} \begin{array}{rcl} \end{array}$ ².

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