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Chapter 11: Q. 33 (page 872)

Find the velocity and acceleration vectors for the position vectors given in Exercises 30鈥�34
$33. r (t) = 鉄� \sin t, \cos t, 2 \sin 2 t 鉄�$

Short Answer

Expert verified

The velocity and acceleration vectors are;

$v (t) = 鉄� \cos t, 鈭� \sin t, 4 \cos 2 t 鉄� a (t) = 鉄� 鈭� \sin t, 鈭� \cos t, 鈭� 8 \sin 2 t 鉄�$

Step by step solution

01

Step 1. Given data

The given position vector is $r (t) = 鉄� \sin t, \cos t, 2 \sin 2 t 鉄�$

We have to find the velocity and acceleration vectors.

02

Step 2. Velocity vector

The velocity vector v(t) is given by

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

Therefore,

$\begin{aligned} v (t) & = \frac{d}{d t} r (t) \\ = \frac{d}{d t} 鉄� \sin t, \cos t, 2 \sin 2 t 鉄� \\ = 鉄� \frac{d}{d t} \sin t, \frac{d}{d t} \cos t, \frac{d}{d t} 2 \sin 2 t 鉄� \\ = 鉄� \cos t, 鈭� \sin t, 4 \cos 2 t 鉄� \end{aligned}$

Hence the velocity vector $v (t) = 鉄� \cos t, 鈭� \sin t, 4 \cos 2 t 鉄�$

03

Step 3. Acceleration vector

The acceleration vector a(t) is given by

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

Therefore,

$\begin{aligned} a (t) & = \frac{d}{d t} v (t) \\ = \frac{d}{d t} 鉄� \cos t, 鈭� \sin t, 4 \cos 2 t 鉄� \\ = 鉄� \frac{d}{d t} (\cos t), \frac{d}{d t} 鈭� \sin t, \frac{d}{d t} 4 \cos 2 t 鉄� \\ = 鉄� 鈭� \sin t, 鈭� \cos t, 鈭� 8 \sin 2 t 鉄� \end{aligned}$

Hence the acceleration vector $a (t) = 鉄� 鈭� \sin t, 鈭� \cos t, 鈭� 8 \sin 2 t 鉄�$

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Most popular questions from this chapter

$Let t = f (蟿) be a differentiable real - valued function of 蟿, and let r (t) be a differentiable vector function with either two or three components such that f (蟿) is in the domain of r for every value of 蟿 on some interval I . Find \frac{d (r (f (蟿)))}{d 蟿} .$

Given a twice-differentiable vector-valued function $r (t)$ and a point $t_{0}$ in its domain, what are the geometric relationships between the unit tangent vector $T (t_{0})$ , the principal unit normal vector $N (t_{0})$ , and ${\frac{d N}{d t}}_{t_{0}}$ ?

As we saw in Example 1, the graph of the vector-valued function $r (t) = (\cos t, \sin t, t), for t 鈭� [0, 2 蟺]$ is a circular helix that spirals counterclockwise around the z-axis and climbs ast increases. Find another parametrization for this helix so that the motion along the helix is faster for a given change in the parameter.

Given a twice-differentiable vector-valued function $r (t)$ , what is the definition of the binormal vector $B (t)$ ?

Given a differentiable vector-valued function $r (t)$ , what is the relationship between $r' (t_{0})$ and $T (t_{0})$ at a point $t_{0}$ in the domain of $r (t)$ ?

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