Chapter 11: Q. 10 (page 901)

Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.
$r (t) = \{t, 2 t^{2}, 3 t^{3}\}$

Short Answer

Expert verified

Ans: $v (t) = <1, 4 t, 9 t^{2}> and a (t) = ⟨ 0, 4, 18 t ⟩$

Step by step solution

Step 1. Given information:

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

Step 2. Denoting the velocity and acceleration of the vector:

The velocity vector, $v (t)$ is given by

$v (t) = \frac{d}{d t} r (t) = <\frac{d}{d t} x (t), \frac{d}{d t} y (t), \frac{d}{d t} z (t)>$ .

That is, we take the derivative of each component function of $r (t)$ .

The acceleration vector, $a (t)$ is given by

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

That is, $a (t)$ is obtained by taking the derivative of each component function of $v (t)$ .

Step 3. Finding the velocity and acceleration of the vector:

Consider

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

$v (t) = \frac{d}{d t} (r (t)) = \frac{d}{d t} <t, 2 t^{2}, 3 t^{3}> = <\frac{d}{d t} (t), \frac{d}{d t} (2 t^{2}), \frac{d}{d t} (3 t^{3})> = <1, 4 t, 9 t^{2}> a (t) = \frac{d}{d t} (v (t)) = \frac{d}{d t} <1, 4 t, 9 t^{2}> = <\frac{d}{d t} (1), \frac{d}{d t} (4 t), \frac{d}{d t} (9 t^{2})> = ⟨ 0, 4, 18 t ⟩$

thus

$v (t) = <1, 4 t, 9 t^{2}> and a (t) = ⟨ 0, 4, 18 t ⟩$

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Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.
$r (t) = \{t, 2 t^{2}, 3 t^{3}\}$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Denoting the velocity and acceleration of the vector:

Step 3. Finding the velocity and acceleration of the vector:

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Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.r(t)=t,2t2,3t3

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Denoting the velocity and acceleration of the vector:

Step 3. Finding the velocity and acceleration of the vector:

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Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.
$r (t) = \{t, 2 t^{2}, 3 t^{3}\}$