Chapter 11: Q. 30 (page 872)

Find the velocity and acceleration vectors for the position vectors given in Exercises 30–34
$30. r (t) = ⟨ t, t^{2}, t^{3} ⟩$

Short Answer

Expert verified

The velocity and acceleration vectors are;

$v (t) = ⟨ 1, 2 t, 3 t^{2} ⟩ a (t) = ⟨ 0, 2, 6 t ⟩$

Step by step solution

Step 1. Given data

The given position vector is $r (t) = ⟨ t, t^{2}, t^{3} ⟩$

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,

$\begin{aligned} v (t) & = \frac{d}{d t} r (t) \\ = \frac{d}{d t} ⟨ t, t^{2}, t^{3} ⟩ \\ = ⟨ \frac{d}{d t} (t), \frac{d}{d t} (t^{2}), \frac{d}{d t} (t^{3}) ⟩ \\ = ⟨ 1, 2 t, 3 t^{2} ⟩ \end{aligned}$

Hence the velocity vector $v (t) = ⟨ 1, 2 t, 3 t^{2} ⟩ \begin{aligned} \end{aligned}$

Step 3. Acceleration vector

The acceleration vector a(t) is given by,

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

Therefore,

$\begin{aligned} a (t) & = \frac{d}{d t} v (t) \\ = \frac{d}{d t} ⟨ 1, 2 t, 3 t^{2} ⟩ \\ = ⟨ \frac{d}{d t} (1), \frac{d}{d t} (2 t), \frac{d}{d t} (3 t^{2}) ⟩ \\ = ⟨ 0, 2, 6 t ⟩ \end{aligned}$

Hence, the acceleration vector is $a (t) = ⟨ 0, 2, 6 t ⟩$

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Find the velocity and acceleration vectors for the position vectors given in Exercises 30–34
$30. r (t) = ⟨ t, t^{2}, t^{3} ⟩$

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 30–3430.r(t)=⟨t,t2,t3⟩

Short Answer

Step by step solution

Step 1. Given data

Step 2. Velocity vector

Step 3. Acceleration vector

One App. One Place for Learning.

Most popular questions from this chapter

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Find the velocity and acceleration vectors for the position vectors given in Exercises 30–34
$30. r (t) = ⟨ t, t^{2}, t^{3} ⟩$