Chapter 11: Q. 11 (page 901)

Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.
$r (t) = (\cos 2 t, \sin 3 t)$

Short Answer

Expert verified

Ans: $v (t) = ⟨ - 2 \sin 2 t, 3 \cos 3 t ⟩ and a (t) = ⟨ - 4 \cos 2 t, - 9 \sin 3 t ⟩$

Step by step solution

Step 1. Given information:

$r (t) = (\cos 2 t, \sin 3 t)$

Step 2. Denoting the velocity and acceleration of the vector:

The velocity vector, $v (t)$ is given by

$v (t) = \frac{d}{d t} r (t) = <\frac{d}{d t} x (t), \frac{d}{d t} y (t), \frac{d}{d t} z (t)>$ .

That is, we take the derivative of each component function of $r (t)$ .

The acceleration vector, $a (t)$ is given by

$a (t) = \frac{d}{d t} v (t)$ .

That is, $a (t)$ is obtained by taking the derivative of each component function of $v (t)$ .

Step 3. Finding the velocity and acceleration of the vector:

Consider $r (t) = ⟨ \cos 2 t, \sin 3 t ⟩$

$v (t) = \frac{d}{d t} r (t) = \frac{d}{d t} ⟨ \cos 2 t, \sin 3 t ⟩ = <\frac{d}{d t} (\cos 2 s t, \sin 3 t)> = <\frac{d}{d t} (\cos 2 t, \frac{d}{d t} (\sin 3 t))> = ⟨ - 2 \sin 2 t, 3 \cos 3 t ⟩ a (t) = \frac{d}{d t} (v (t)) = \frac{d}{d t} ⟨ - 2 \sin 2 t, 3 \cos 3 t ⟩ = <\frac{d}{d t} (- 2 \sin 2 t), \frac{d}{d t} (3 \cos 3 t)> = ⟨ - 4 \cos 2 t, - 9 \sin 3 t ⟩$

thus

$v (t) = ⟨ - 2 \sin 2 t, 3 \cos 3 t ⟩ and a (t) = ⟨ - 4 \cos 2 t, - 9 \sin 3 t ⟩$

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Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.
$r (t) = (\cos 2 t, \sin 3 t)$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Denoting the velocity and acceleration of the vector:

Step 3. Finding the velocity and acceleration of the vector:

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Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.r(t)=(cos2t,sin3t)

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Denoting the velocity and acceleration of the vector:

Step 3. Finding the velocity and acceleration of the vector:

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Most popular questions from this chapter

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Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.
$r (t) = (\cos 2 t, \sin 3 t)$