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Chapter 11: Q. 18 (page 898)

$r (t) = (t, t^{2})$

Short Answer

Expert verified

Thus $a_{T} = \frac{4 t}{\sqrt{1 + 4 t^{2}}}$ and $a_{N} = \frac{2}{\sqrt{1 + 4 t^{2}}}$

Step by step solution

01

Introduction

Consider $r (t) = <t, t^{2}>$

The goal is to determine the tangential and normal components of acceleration. $r (t)$ .

02

Given information

The tangential and normal components of Acceleration

$r (t)$ is a twice - differentiable vector function with $r^{'} (t) = v (t)$ and $r^{''} (t) = a (t)$ .

The tangential component of acceleration, $a_{T} = \frac{v 路 a}{鈥� v 鈥�}$ . The normal component acceleration,

a_{N} = \frac{鈥� v 脳 a 鈥�}{鈥� v 鈥�} r (t) = <t, t^{2}> v (t) = r^{'} (t) = 鉄� 1, 2 t 鉄� 鈥� v 鈥� = 鈥� 鉄� 1, 2 t 鉄� 鈥� = \sqrt{1 + 4 t^{2}} v 路 a = v (t) 路 a (t) = 鉄� 1, 2 t 鉄� 路 鉄� 0, 2 鉄� = 1 (0) + 2 t (2) = 4 t

03

Explanation

$v 脳 a = v (t) 脳 a (t)$

$= |\begin{matrix} i j k \\ 12 t 0 \\ 0 & 20 \end{matrix}|$

$= i (0) - j (0) + k (2)$

$= 鉄� 0, 0, 2 鉄�$

$鈥� v 脳 a 鈥� = \sqrt{2^{2}} = 2$

The tangential component of acceleration,

a_{T} = \frac{v 路 a}{鈥� v 鈥�} a_{r} = \frac{4 t}{\sqrt{1 + 4 t^{2}}}

The normal component of acceleration,

a_{N} = \frac{鈥� v 脳 a 鈥�}{鈥� v 鈥�} = \frac{2}{\sqrt{1 + 4 t^{2}}} Thus a_{T} = \frac{4 t}{\sqrt{1 + 4 t^{2}}} and a_{N} = \frac{2}{\sqrt{1 + 4 t^{2}}}

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Most popular questions from this chapter

$Find the derivative of the vector - valued function r (t) = t^{2} i 鈭� 6 t^{2} j, t > 0, and show that the derivative at any point t_{0} is a scalar multiple of the derivative at any other point . What does this property say about the graph of r ? Use that information to sketch the graph of r .$

For each of the vector-valued functions, find the unit tangent vector.

$r (t) = (t, t^{2}, t^{3})$

Find parametric equations for each of the vector-valued functions in Exercises 26鈥�34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r (t) = (1 + \sin t, 3 - \cos 2 t), f o r t 鈭� [0, 2 蟺]$

Evaluate and simplify the indicated quantities in Exercises 35鈥�41.
$t (\sin t, \cos t)$

$Complete the following definition : If r (t) = <x (t), y (t), z (t)> is a differentiable position function, then the velocity vector v (t) is . . . .$

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