Chapter 11: Q. 32 (page 872)

Find the velocity and acceleration vectors for the position vectors given in Exercises 30–34
$32 . r (t) = ⟨ \sec t, \frac{1}{t}, e^{t} \ln t ⟩$

Short Answer

Expert verified

The velocity and acceleration vectors are;

$\begin{aligned} v (t) = ⟨ \sec t \tan t, \frac{- 1}{t^{2}}, e^{t} (\frac{1}{t} + \ln t) ⟩ \\ a (t) = ⟨ \sec^{3} + \sec t \tan^{2} t, \frac{2}{t^{3}}, e^{t} (\frac{- 1}{t^{2}} + \frac{2}{t} + \ln t) ⟩ \end{aligned}$

Step by step solution

Step 1. Given data

The given position vector is $r (t) = ⟨ \sec t, \frac{1}{t}, e^{t} \ln t ⟩$

We have to find the velocity and acceleration vectors.

Step 2. Velocity vector

The velocity vector v(t) is given by

$v (t) = \frac{d}{d t} r (t) = ⟨ \frac{d}{d t} x (t), \frac{d}{d t} y (t), \frac{d}{d t} z (t) ⟩$

Therefore,

$\begin{aligned} v (t) & = \frac{d}{d t} r (t) \\ = \frac{d}{d t} ⟨ \sec t, \frac{1}{t}, e^{t} \ln t ⟩ \\ = ⟨ \frac{d}{d t} (\sec t), \frac{d}{d t} \frac{1}{t}, \frac{d}{d t} e^{t} \ln t ⟩ \\ = ⟨ \sec t \tan t, \frac{- 1}{t^{2}}, \frac{e^{t}}{t} (\ln t) (e^{t}) ⟩ \\ = ⟨ \sec t \tan t, \frac{- 1}{t^{2}}, e^{t} (\frac{1}{t} + \ln t) ⟩ \end{aligned}$

Hence the velocity vector $v (t) = ⟨ \sec t \tan t, \frac{- 1}{t^{2}}, e^{t} (\frac{1}{t} + \ln t) ⟩$

Step 3. Acceleration vector

The acceleration vector a(t) is given by

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

Therefore,

$\begin{aligned} a (t) & = \frac{d}{d t} v (t) \\ = \frac{d}{d t} ⟨ \sec t \tan t, \frac{- 1}{t^{2}}, e^{t} (\frac{1}{t} + \ln t) ⟩ \\ = ⟨ \sec^{3} t + \sec t \tan^{2} t, \frac{2}{t^{3}}, e^{t} (\frac{- 1}{t^{2}} + \frac{1}{t}) + (\frac{1}{t} + \ln t) e^{t} ⟩ \\ = ⟨ \sec^{3} t + \sec t \tan^{2} t, \frac{2}{t^{3}}, e^{t} (\frac{- 1}{t^{2}} + \frac{2}{t} + \ln t) ⟩ \end{aligned}$

Hence the acceleration vector $a (t) = ⟨ \sec^{3} t + \sec t \tan^{2} t, \frac{2}{t^{3}}, e^{t} (\frac{- 1}{t^{2}} + \frac{2}{t} + \ln t) ⟩$

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Find the velocity and acceleration vectors for the position vectors given in Exercises 30–34
$32 . r (t) = ⟨ \sec t, \frac{1}{t}, e^{t} \ln t ⟩$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Velocity vector

Step 3. Acceleration vector

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Find the velocity and acceleration vectors for the position vectors given in Exercises 30–34
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