Question

A particle's motion is described by the following two parametric equations: $$ \begin{array}{l} x(t)=5 \cos (2 \pi t) \\ y(t)=5 \sin (2 \pi t) \end{array} $$ where the displacements are in meters and \(t\) is the time, in seconds. a) Draw a graph of the particle's trajectory (that is, a graph of \(y\) versus \(x\) ). b) Determine the equations that describe the \(x\) - and \(y\) -components of the velocity, \(v_{x}\) and \(v_{y}\), as functions of time. c) Draw a graph of the particle's speed as a function of time.

Short answer

In this exercise, we looked at the motion of a particle described by the parametric equations \(x(t) = 5\cos(2\pi t)\) and \(y(t) = 5\sin(2\pi t)\). By examining the particle's trajectory, we found that it follows a circular path with a radius of 5 meters and period of 1 second. We differentiated the parametric equations to determine the x- and y-components of velocity, which are given by \(v_x(t) = -10\pi\sin(2\pi t)\) and \(v_y(t) = 10\pi\cos(2\pi t)\). Finally, we calculated the particle's speed, which is constant at 10π meters per second.

Step-by-step solution

Convert Parametric Equations to Position Vector

Combine the given parametric equations by considering them as a position vector \(\textbf{r}(t)\): $$ \textbf{r}(t) = x(t)\textbf{i} + y(t)\textbf{j} = 5\cos(2\pi t)\textbf{i} + 5\sin(2\pi t)\textbf{j}. $$

Eliminate Parameter t

Divide the y-coordinate equation by the x-coordinate equation to eliminate the parameter t: $$ \frac{y(t)}{x(t)} = \frac{5\sin(2\pi t)}{5\cos(2\pi t)} = \tan(2\pi t). $$ From the above equation, we have: $$ 2\pi t = \arctan\left(\frac{y(t)}{x(t)}\right). $$

Draw the Graph

From the given parametric equations, we have the following relations: $$ \begin{cases} x(t) = 5\cos(2\pi t)\\ y(t) = 5\sin(2\pi t) \end{cases}. $$ This represents a circle with the radius 5 meters and a period of 1 second. Plot the trajectory on the xy-plane by eliminating t using the above equation. #b) Equations for x- and y-components of Velocity#

Differentiate Parametric Equations

To find the x- and y-components of the velocity, differentiate the given parametric equations with respect to time t: $$ v_x(t) = \frac{dx(t)}{dt} = -10\pi\sin(2\pi t) $$ and $$ v_y(t) = \frac{dy(t)}{dt} = 10\pi\cos(2\pi t). $$ These equations describe the x- and y-components of the velocity as functions of time. #c) Graph of Particle's Speed as a Function of Time#

Calculate the Speed

The speed (magnitude of the velocity vector) can be calculated as: $$ v(t) = \sqrt{v_x(t)^2 + v_y(t)^2} = \sqrt{(-10\pi\sin(2\pi t))^2 + (10\pi\cos(2\pi t))^2}. $$ After simplifying the expression, we get: $$ v(t) = 10\pi. $$

Draw the Graph

From the above calculation, we can see that the speed is constant over time and is independent of the time t. Thus, the graph of the particle's speed as a function of time will be a horizontal line at the value \(v(t) = 10\pi\) meters per second.