Q. 34 Find the velocity and accelerati... [FREE SOLUTION]

Chapter 11: Q. 34 (page 872)

Find the velocity and acceleration vectors for the position vectors given in Exercises 30鈥�34
$r (t) = \cos 3 t i + \sin 4 t j + t k$

Short Answer

Expert verified

The velocity and acceleration vectors are:

$\begin{aligned} v (t) = 鉄� 鈭� 3 s i n 3 t, 4 c o s 4 t, 1 鉄� \\ a (t) = 鉄� 鈭� 9 c o s 3 t 鈭� 16 s i n 4 t, 0 鉄� \end{aligned}$

Step by step solution

Step 1. Given data

The given position vector is $r (t) = \cos 3 t i + \sin 4 t j + t k$

We have to find the velocity and acceleration vectors.

Step 2. Velocity vector

The velocity vector v(t) is given by

$v (t) = \frac{d}{d t} r (t) = x^{鈥�} (t) i + y^{鈥�} (t) j + z^{鈥�} (t) k = 鉄� x^{鈥�} (t), y^{鈥�} (t), z^{鈥�} (t) 鉄�$

Therefore,

$\begin{aligned} v (t) & = \frac{d}{d t} (r (t)) \\ = \frac{d}{d t} (\cos 3 t i + \sin 4 t j + t k) \\ = \frac{d}{d t} (\cos 3 t) i + \frac{d}{d t} (\sin 4 t) j + \frac{d}{d t} (t) k \\ = 鈭� 3 \sin t i + 4 \cos 4 t j + k \\ = 鉄� 鈭� 3 \sin 3 t, 4 \cos 4 t, 1 鉄� \end{aligned}$

Hence the velocity vector $v (t) = 鉄� 鈭� 3 s i n 3 t, 4 c o s 4 t, 1 鉄�$ .

Step 3. Acceleration vector

The acceleration vector a(t) is given by

$a (t) = \frac{d}{d t} v (t) = x^{鈥测赌�} (t) i + y^{鈥测赌�} (t) j + z " (t) k = 鉄� x^{鈥测赌�} (t), y^{鈥测赌�} (t), z^{鈥测赌�} (t) 鉄�$

Therefore,

$\begin{aligned} a (t) & = \frac{d}{d t} (v (t)) \\ = \frac{d}{d t} 鉄� 鈭� 3 \sin 3 t, 4 \cos 4 t, 1 鉄� \\ = 鉄� \frac{d}{d t} (鈭� 3 \sin 3 t), \frac{d}{d t} (4 \cos 4 t), \frac{d}{d t} (1) 鉄� \\ = 鉄� 鈭� 9 \cos 3 t, 鈭� 16 \sin 4 t, 0 鉄� \end{aligned}$

Hence the acceleration vector $a (t) = 鉄� 鈭� 9 c o s 3 t 鈭� 16 s i n 4 t, 0 鉄�$

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91影视

Find the velocity and acceleration vectors for the position vectors given in Exercises 30鈥�34
$r (t) = \cos 3 t i + \sin 4 t j + t k$

Short Answer

Step by step solution

Step 1. Given data

Step 2. Velocity vector

Step 3. Acceleration vector

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91影视

Find the velocity and acceleration vectors for the position vectors given in Exercises 30鈥�34r(t)=cos3ti+sin4tj+tk

Short Answer

Step by step solution

Step 1. Given data

Step 2. Velocity vector

Step 3. Acceleration vector

One App. One Place for Learning.

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Find the velocity and acceleration vectors for the position vectors given in Exercises 30鈥�34
$r (t) = \cos 3 t i + \sin 4 t j + t k$