Chapter 4.3, Problem 31E

(a)

To determine

To prove:That the particle is always moving towards the right.

Explanation of Solution

Given information:A particle is moving along x- axis and its position is given by,

  $x(t)=\arctan \left(t\right)$ .

  $t\ge 0$

Formula used:

  $v=\frac{dx}{dt}$........(1)

Where, v = velocity of object

x = position of object at time t

Proof: Using the formula given in equation (1),

  $\begin{array}{l}v=\frac{dx}{dt}\\ v=\frac{d\left({\tan }^{-1}\left(t\right)\right)}{dt}\\ v=\frac{1}{1+t^2}\end{array}$

At any time t the velocity of the particle is positive,

So it proves that the particle is always moving to right.

(b)

To determine

To prove:That the particle is continuously decelerating.

Explanation of Solution

Formula used:

  $a=\frac{dv}{dt}$........(2)

Where, a = acceleration at time t

v = velocity of object at time t.

Proof: Using the formula given in equation (2),

  $\begin{array}{l}a=\frac{dv}{dt}\\ a=\frac{d\left(\frac{1}{1+t^2}\right)}{dt}\\ a=\frac{-1}{{\left(1+t^2\right)}^2}\times \frac{d\left(1+t^2\right)}{dt}\\ a=\frac{-2t}{1+t^4+2t^2}\end{array}$

At any time t the velocity of the particle is positive,

So it proves that the particle is continuously decelerating.

(c)

To determine

To calculate:The position of particle as t tends to infinity.

Answer to Problem 31E

The position of particle is $\frac{\pi }{2}$ .

Explanation of Solution

Calculation:Calculate the position as t tends to infinity,

  $\begin{array}{l}x=\underset{t\to \infty }{\lim }\left(\arctan \left(t\right)\right)\\ x=\arctan \left(\infty \right)\\ x=\frac{\pi }{2}\end{array}$

So the position of particle is $\frac{\pi }{2}$ .