Chapter 10.2, Problem 40E

To determine

To find: The position of the particle at time $t=3$ .

Answer to Problem 40E

The answer: The position of the particle at time $t=3$ is equal to $(7,2)$

Explanation of Solution

Given information:

The velocity of $v(t)=(2\pi \cos 4\pi t,4\pi \sin 2\pi t);(7,2)$

Calculation:

Know that the position vector of a particle at time $b$ is equal to the position vector of a particle at time $a$ plus the displacement of a particle from $t=a$

To

  $t=b$ .

Know that,

  $v(t)=v_1(t),v_2(t)$

Then,

  

In this exercise,

  $v(t)=(2\pi \cos 4\pi t,4\pi \sin 2\pi t)$

Then,

  $\begin{array}{c}{v}_1(t)=2\pi \cos (4\pi t)\\ v_2(t)=4\pi \sin (2\pi t)\end{array}$

Since,

  

To calculate the integral:

  $\begin{array}{c}{{\displaystyle \int }}^{\text{}}\cos (4\pi t)dt=\frac{1}{4\pi }{{\displaystyle \int }}^{\text{}}\cos udu\\ =\frac{\sin u}{4\pi }+C\\ =\frac{\sin (4\pi t)}{4\pi }+C\end{array}$

To use the previous results, the displacement of the particle from $t=0$ to

  $t=3$ is equal to $(0,0)$ .

The position of the particle at time $t=0$ is equal to $(7,2)$ , which means that,

  $r(0)=(7,2)$ .

Then,

  $\begin{array}{c}r(3)=(7,2)+(0,0)\\ =(7,2)\end{array}$

Therefore, the required position of the particle at time $t=3$ is equal to $(7,2)$ .

$\text{(b)}$

To determine

To find: The distance the particle travels from $t=0$ to

  $t=3$ .

Answer to Problem 40E

The answer: The distance the particle travels from $t=0$ to

  $t=3$ is approximately equal to $28.523$ .

Explanation of Solution

Given information:

The velocity of $v(t)=(2\pi \cos 4\pi t,4\pi \sin 2\pi t);(7,2)$ .

Calculation:

Know that the position vector of a particle at time $b$ is equal to the position vector of a particle at time $a$ plus the displacement of a particle from $t=a$

To

  $t=b$ .

Know that,

  $v(t)=v_1(t),v_2(t)$

Is

  

In this exercise,

  $v(t)=(2\pi \cos 4\pi t,4\pi \sin 2\pi t)$

Then,

  $\begin{array}{c}{v}_1(t)=2\pi \cos (4\pi t)\\ v_2(t)=4\pi \sin (2\pi t)\end{array}$

Since, The distance the particle travels from $t=0$ to

  $t=3$ .

Is equal to:

  

Therefore, the required distance the particle travels from $t=0$ to

  $t=3$ is approximately equal to $28.523$ .