Chapter 11: Q.48 (page 872)

Use the given velocity vectors $v (t) = r' (t)$ and initial positions in exercise to find the position function $r (t)$ .
localid="1650740363372" $v (t) = \cos t i + \sin t j + t k, r (0) = - i + 5 j$

Short Answer

Expert verified

$r (t) = (sint - 1) i + (- cost + 6) j + \frac{t^{2}}{2} k$

Step by step solution

Given Information

Consider $v (t) = \cos t i + \sin t j + t k, r (0) = - i + 5 j$

The objective is to find the position function $r (t)$ .

Since $v (t) = r' (t)$

We have,

$r (t) = \int v (t) d t = \int (costi + sintj + tk) dt = i \int \cos t d t + j \int \sin t d t + k \int t d t = \sin t i - \cos t j + \frac{t^{2}}{2} k + c = (\sin t + c_{1}) i + (- \cos t + c_{2}) j + (\frac{t^{2}}{2} + c_{3}) k$

Where $c_{1}, c_{2}$ and $c_{3}$ are scalar constants.

Calculation

$r (t) = (\sin t + c_{1}) i + (- \cos t + c_{2}) j + (\frac{t^{2}}{2} + c_{3}) k r (0) = (\sin 0 + c_{1}) i + (- \cos 0 + c_{2}) j + (\frac{0^{2}}{2} + c_{3}) k r (0) = c_{1} i + (- 1 + c_{2}) j + c_{3} k . . . . . . . . . . . . . . (1) r (0) = - i + 5 j . . . . . . . . . . . . . . . . (2)$

comparing (1) and (2) equations,

$c 1 = - 1 . - 1 + c 2 = 5, c 3 = 0 c 2 = 6$

$r (t) = (sint - 1) + (- cost + 6) j + (\frac{t^{2}}{2}) k$

Calculation

Check: Taking the derivatives of $r (t)$ , we obtain

$r^{2} (t) = \cos t i + \sin t j + t k,$

Which is the correct tangent vector function

Next we evaluate

$r (0) = (\sin 0 - 1) i + (- \cos 0 + 6) j + (\frac{0^{2}}{2}) k = - i + 5 j$

Which is true according to the problem

Thus $r (t) = (\sin t - 1) i + (- \cos t + 6) j + \frac{t^{2}}{2} k$

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Use the given velocity vectors $v (t) = r' (t)$ and initial positions in exercise to find the position function $r (t)$ .
localid="1650740363372" $v (t) = \cos t i + \sin t j + t k, r (0) = - i + 5 j$

Short Answer

Step by step solution

Given Information

Calculation

Calculation

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Use the given velocity vectors v(t)=r'(t)and initial positions in exercise to find the position function r(t).localid="1650740363372" v(t)=costi+sintj+tk,r(0)=-i+5j

Short Answer

Step by step solution

Given Information

Calculation

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Theoretical and Mathematical Physics

Applied Mathematics

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Use the given velocity vectors $v (t) = r' (t)$ and initial positions in exercise to find the position function $r (t)$ .
localid="1650740363372" $v (t) = \cos t i + \sin t j + t k, r (0) = - i + 5 j$