Question

A race Two people travel from \(P(4,0)\) to \(Q(-4,0)\) along the paths given by $$\begin{aligned} \mathbf{r}(t) &=\left\langle 4 \cos \frac{\pi t}{8}, 4 \sin \frac{\pi t}{8}\right\rangle \text { and } \\ \mathbf{R}(t) &=\left\langle 4-t,(4-t)^{2}-16\right\rangle \end{aligned}$$ a. Graph both paths between \(P\) and \(Q\). b. Graph the speeds of both people between \(P\) and \(Q\). c. Who arrives at \(Q\) first?

Short answer

Based on the given parametric equations, we analyzed the paths taken by two people from point P to point Q by graphing their paths, finding their speeds, and determining the time taken to reach point Q. Both people take the same amount of time (t=8) to reach point Q, so they both arrive at Q at the same time.

Step-by-step solution

Graph the paths between P and Q

First, we need to graph the parametric equations to visualize the paths: For the first person, the path is described by: $$\mathbf{r}(t) =\begin{bmatrix}4 \cos(\frac{\pi t}{8})\\ 4 \sin(\frac{\pi t}{8})\end{bmatrix}$$ To graph this, consider the interval \(0 \leq t \leq 8\), as the person starts at \(P(4,0)\) when \(t=0\) and reaches \(Q(-4,0)\) when \(t=8\). For the second person, the path is described by: $$\mathbf{R}(t) =\begin{bmatrix}4-t\\ (4-t)^{2}- 16\end{bmatrix}$$ To graph this, consider the interval \(0 \leq t \leq 8\), as the person starts at \(P(4,0)\) when \(t=0\) and reaches \(Q(-4,0)\) when \(t=8\). Plot both paths on the same graph to visualize them.

Find the speed of both people

Now, we will find the speed of both people by taking the derivative of their position vectors: For the first person, differentiate \(\mathbf{r}(t)\) with respect to \(t\): $$\mathbf{r'}(t) = \begin{bmatrix} -\frac{\pi}{2}\sin(\frac{\pi t}{8})\\ \frac{\pi}{2}\cos(\frac{\pi t}{8})\end{bmatrix}$$ Then find the magnitude, which represents the speed: $$v(t) = \sqrt{(-\frac{\pi}{2}\sin(\frac{\pi t}{8}))^2 + (\frac{\pi}{2}\cos(\frac{\pi t}{8}))^2}$$ For the second person, differentiate \(\mathbf{R}(t)\) with respect to \(t\): $$\mathbf{R'}(t) = \begin{bmatrix}-1\\ -2(4-t)\end{bmatrix}$$ Then find the magnitude, which represents the speed: $$V(t) = \sqrt{(-1)^2 + (-2(4-t))^2}$$

Graph the speeds of both people

Now, plot the speed functions \(v(t)\) and \(V(t)\) on a graph for the interval \(0 \leq t \leq 8\).

Determine who arrives at Q first

To find the time it takes for each person to reach point Q, we need to find the value of \(t\) when they are at point \(Q(-4,0)\). For the first person: Set \(x = -4\) in \(\mathbf{r}(t)\) and solve for \(t\): $$-4 = 4 \cos(\frac{\pi t}{8})$$ which gives \(t=8\). For the second person: Set \(x = -4\) in \(\mathbf{R}(t)\) and solve for \(t\): $$-4 = 4 - t$$ which gives \(t=8\). Since both people take the same amount of time (t=8) to reach point Q, they both arrive at Q at the same time.

Key concepts

Graphing Parametric Paths

Visualizing the trajectory of an object in motion is essential in understanding its behavior. When working with parametric equations, we're given a set of functions that describe the X and Y coordinates as functions of a third variable, typically time (t). In the exercise provided, the paths of two people traveling between two points are described by parametric equations.

To graph these paths, we plot the parametric equations on a coordinate system. For the first person, their path is a segment of a circle, as indicated by their equations involving sine and cosine functions. The equations for the second person suggest a parabolic path. When plotting, it's important to consider the range of the independent variable t, ensuring it reflects the journey from start to end. The interval given in the exercise, from 0 to 8, captures the complete journey for both individuals.

By graphing these parametric paths, students can visually compare the trajectories and get a clear picture of the motion along paths that are not simply straight lines. This graph reinforces understanding how changing the parameter, t, affects the position of the moving point, thus influencing the shape of the path taken.

Calculating Speed from Parametric Equations

Speed, a scalar quantity representing how fast an object is moving, can be calculated from parametric equations by finding the derivative of the position vectors with respect to time. This differentiation yields the velocity vector, whose magnitude is the speed.

For instance, in the exercise, we first differentiate the position vector of each person with respect to t, yielding their respective velocity vectors. To find the speed, we calculate the magnitude of these velocity vectors using the Pythagorean theorem: the square root of the sum of the squared components.

For the circular path, the speed is constant because the radius of the circle does not change, as evidenced by the unchanging magnitude of its derivative. Conversely, for the parabolic path, the magnitude of the derivative changes with t, indicating that the speed varies over time. Therefore, graphing the speed against time for both paths would show a constant line for one, and a line that changes in intensity for the other, visually conveying differences in their motions.

Determining Arrival Time with Parametric Functions

When studying motion, it's often important to determine when an object will arrive at a particular location. Parametric functions provide a robust framework for tackling such problems since they describe the position of the object at any given time.

In the provided exercise, we aim to compare the arrival times of two people traveling along different paths to the same destination. To do this, we set the x-component of each person's position equal to the x-coordinate of the destination and solve for t. This gives us the precise moment in time when each person reaches the endpoint.

Surprisingly, despite the differing paths, both individuals arrive simultaneously, which could lead to questions about the average speed and acceleration over the journey. This result teaches students that different paths can result in the same travel time and that parametric functions are a powerful tool in analyzing motion along a specific path.