Chapter 11: Q.49 (page 872)

Use the given acceleration vectors $a (t) = v (t) = r (t)$ and initial conditions in Exercises to find the position function $r (t)$ .
localid="1650737775645" $a (t) = 2 t, 3 t^{2}, v (0) = 1, - 2, r (0) = 2, - 3$

Short Answer

Expert verified

$r (t) = (\frac{r^{2}}{3} + t + 2 \frac{t^{2}}{4} - 2 i - 3)$

$r (t) = (\frac{t^{3}}{3} + t + 2, \frac{t^{4}}{4} - 2 t - 3)$

Step by step solution

Given Information

Consider $a (t) = (2 t, 3 t^{2}), v (0) = (1, - 2), r (0) = (2, - 3)$

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

Since $a (t) = v^{t} (t) = r^{n} (t)$ , we have $v (t) = \int a (t) dt and r (t) = \int v (t) dt$

Now,

But the given initial position $v (0) = (1, - 2) = i - 2 j . . . . . . . . . . . . . . (2)$

Comparing (1) and (2) equations, $C_{1} = 1, C_{2} = - 2$

Calculation

$v (t) = (t^{2} + 1) i + (t^{3} - 2) j$

Now,

$r (t) = \int v (t 0 d t = \int ((t^{2} + 1) i + (t^{2} - 2) j) d t$

$= (\frac{t^{3}}{3} + t) i + (\frac{t^{4}}{4} - 2 t) j + c$

Where c is a vector constant

$= (\frac{t^{3}}{3} + t + c_{3}) i + (\frac{t^{4}}{4} - 2 t + c_{4}) j$

where $c_{3}$ and $c_{4}$ are scalars.

$r (0) = (\frac{0^{3}}{3} + 0 + c_{3}) i + (\frac{0^{4}}{4} - 2 (0) + c_{4}) j r (0) = c_{3} i + c_{4} j . . . . . . . . . . (3)$

But as per problem, $r (0) = (2, - 3) = 2 i - 3 j . . . . . . . . . . . (4)$

Comparing $(3) & (4)$ equations, $c_{3} = 2, c_{4} = - 3$

Thus

$r (t) = (\frac{t^{3}}{3} + t + 2) i + (\frac{t^{4}}{4} - 2 t - 3) j = (\frac{t^{3}}{3} + t + 2, \frac{t^{4}}{4} - 2 t - 3)$

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Use the given acceleration vectors $a (t) = v (t) = r (t)$ and initial conditions in Exercises to find the position function $r (t)$ .
localid="1650737775645" $a (t) = 2 t, 3 t^{2}, v (0) = 1, - 2, r (0) = 2, - 3$

Short Answer

Step by step solution

Given Information

Calculation

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Use the given acceleration vectors a(t)=v(t)=r(t)and initial conditions in Exercises to find the position function r(t).localid="1650737775645" a(t)=2t,3t2,v(0)=1,−2,r(0)=2,−3

Short Answer

Step by step solution

Given Information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Applied Mathematics

Statistics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Use the given acceleration vectors $a (t) = v (t) = r (t)$ and initial conditions in Exercises to find the position function $r (t)$ .
localid="1650737775645" $a (t) = 2 t, 3 t^{2}, v (0) = 1, - 2, r (0) = 2, - 3$