Chapter 11: Q. 1TF (page 891)

A decomposition of the acceleration vector: Find ${comp}_{v (t)} a (t),$ wherev and a are the velocity and acceleration vectors, respectively, of the following functions.
$r (t) = <t, t^{2}, t^{3}>$

Short Answer

Expert verified

The component of a(t)along v(t)is ${comp}_{v (t)} a (t) = \frac{4 t + 18 t^{3}}{\sqrt{1 + 4 t^{2} + 9 t^{4}}} .$

Step by step solution

Step 1. Given Information.

The given function is $r (t) = <t, t^{2}, t^{3}> .$

Step 2. Calculating the velocity and acceleration vector.

Let's calculate the given function,

$r (t) = <t, t^{2}, t^{3}> r' (t) = v (t) = <1, 2 t, 3 t^{2}> r'' (t) = a (t) = <0, 2, 6 t> Now, ||v (t)|| = \sqrt{{(1)}^{2} + {(2 t)}^{2} + {(3 t^{2})}^{2}} = \sqrt{1 + 4 t^{2} + 9 t^{4}} . . . . . . . . . . (a) Let' s find (a (t) \cdot v (t)) = (0, 2, 6 t) \cdot (1, 2 t, 3 t^{2}) = 4 t + 18 t^{3} . . . . . . . . . . (b)$

Step 3. Solve.

Put (a) and (b) in $c o m p_{v (t)} a (t) = \frac{|a (t) . v (t)|}{||v (t)||},$

$c o m p_{v (t)} a (t) = \frac{|4 t + 18 t^{3}|}{\sqrt{1 + 4 t^{2} + 9 t^{4}}} c o m p_{v (t)} a (t) = \frac{4 t + 18 t^{3}}{\sqrt{1 + 4 t^{2} + 9 t^{4}}}$

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A decomposition of the acceleration vector: Find ${comp}_{v (t)} a (t),$ wherev and a are the velocity and acceleration vectors, respectively, of the following functions.
$r (t) = <t, t^{2}, t^{3}>$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculating the velocity and acceleration vector.

Step 3. Solve.

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A decomposition of the acceleration vector: Find compv(t)a(t), wherev and a are the velocity and acceleration vectors, respectively, of the following functions.r(t)=t,t2,t3

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Calculating the velocity and acceleration vector.

Step 3. Solve.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Pure Maths

Calculus

Discrete Mathematics

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

A decomposition of the acceleration vector: Find ${comp}_{v (t)} a (t),$ wherev and a are the velocity and acceleration vectors, respectively, of the following functions.
$r (t) = <t, t^{2}, t^{3}>$