Chapter 10.2, Problem 35E

To determine

To find: The velocity and acceleration vector and determine the path of the particle.

Answer to Problem 35E

  $\begin{array}{c}v(t)=(-3\sin (3t),2\cos (2t))\\ a(t)=(-9\cos (3t),-4\sin (2t))\end{array}$

And the path of the particle is the parametric curve defined by:

  $\begin{array}{c}x(t)=\cos (3t)\\ y(t)=\sin (2t)\end{array}$

Explanation of Solution

Given information:

In the plane with position vector $\text{ (}\cos 3t,\sin 2t)$

Calculation:

Know that if the position vector of a particle is equal to:

  $r(t)=\left(x(t),y(t)\right)$

Then the velocity vector of a particle is equal to:

  $v(t)=\left(\frac{dx}{dt},\frac{dy}{dt}\right)$

In this exercise, the position vector of the particle is equal to $\text{ (}\cos 3t,\sin 2t)$

Then,

  $\begin{array}{c}x(t)=\cos (3t)\\ y(t)=\sin (2t)\end{array}$

Which means that,

  $\begin{array}{c}\frac{dx}{dt}=-\sin (3t)\cdot 3\\ =-3\sin (3t)\\ \frac{dy}{dt}=\cos (2t)\cdot 2\\ =2\cos (2t)\end{array}$

The previous results show that the particle's velocity vector is equal to:

  $v(t)=(-3\sin (3t),2\cos (2t))$

Then the acceleration vector of a particle is equal to:

  $a(t)=\left(\frac{d^{2}x}{d{t}^2},\frac{d^{2}y}{d{t}^2}\right)$

Hence, from part $\text{(a)}$ :

  $\begin{array}{c}dx/dt=-3\sin (3t)\\ dy/dt=2\cos (2t)\end{array}$

Then,

  $\begin{array}{c}\frac{d^{2}x}{d{t}^2}=\frac{d(dx/dt)}{dt}\\ =-3\cos (3t)\cdot 3\\ =-9\cos (3t)\\ \frac{d^{2}y}{d{t}^2}=\frac{d(dy/dt)}{dt}\\ =2\cdot (-\sin (2t))\cdot 2\\ =-4\sin (2t)\end{array}$

The previous results show that the particle's acceleration vector is equal to:

  $a(t)=(-9\cos (3t),-4\sin (2t))$

To determine the path of the particle need to graph the parametric curve defined by:

  $\begin{array}{c}x(t)=\cos (3t)\\ y(t)=\sin (2t)\end{array}$

The following graph is the graph of the path of the particle.

  

Therefore, the required velocity vector $v(t)$ is $v(t)=(-3\sin (3t),2\cos (2t))$ and acceleration vector $a(t)$ is $a(t)=(-9\cos (3t),-4\sin (2t))$ .

And the path of the particle is the parametric curve defined by:

  $\begin{array}{c}x(t)=\cos (3t)\\ y(t)=\sin (2t)\end{array}$