Question

A race Two people travel from \(P(4,0)\) to \(Q(-4,0)\) along the paths given by $$ \begin{aligned} \mathbf{r}(t) &=(4 \cos (\pi t / 8), 4 \sin (\pi t / 8)\rangle \text { and } \\\ \mathbf{R}(t) &=\left(4-t,(4-t)^{2}-16\right) \end{aligned} $$ 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

Answer: Both Person 1 and Person 2 arrive at point Q at the same time, taking 8 units of time.

Step-by-step solution

(Determine the total length of each path)

The first path is a semi-circle, with a total radius of 4, so we can calculate its length using the formula \(L = \pi * r = \pi * 4 = 4\pi\). The second path can be obtained by substituting t and rewriting the equation of R(t): $$ \mathbf{R}(t) =\left(4-t,(t-4)^{2}-16\right) = \left(4-t, t^2-8t\right) $$ The length of the second path is a parabola and can't be computed analytically like the first path.

(Graph both paths between points P and Q)

Use a graphing tool (such as Desmos or GeoGebra) to graph the equations of the given paths: 1. Path one: \(\mathbf{r}(t) =(4 \cos (\pi t / 8), 4 \sin (\pi t / 8)\rangle)\) for \(t\) in \([0, 8]\); 2. Path two: \(\mathbf{R}(t) =\left(4-t, t^2-8t\right)\) for \(t\) in \([0, 8]\); You should see a semicircle and a parabolic shape as the two paths from point P(4,0) to Q(-4,0).

(Determine the speed functions for both people)

To find the speed of each person, we need to find the derivatives of their respective paths: 1. For path one, \(\mathbf{r}'(t) = (-\frac{1}{2}\pi\sin(\frac{\pi t}{8}), \frac{1}{2}\pi\cos(\frac{\pi t}{8}))\). The speed function, \(v_1(t) = |\mathbf{r}'(t)| = \frac{1}{2}\pi\) which is a constant. 2. For path two, \(\mathbf{R}'(t) = (-1, 2t - 8)\). The speed function, \(v_2(t) = |\mathbf{R}'(t)| = \sqrt{(-1)^2 + (2t-8)^2}\).

(Graph the speeds of both people between points P and Q)

Graph the speed functions for both people: 1. Speed of Person 1: \(v_1(t) = \frac{1}{2}\pi\); 2. Speed of Person 2: \(v_2(t) = \sqrt{(-1)^2 + (2t-8)^2}\); You will see that Person 1 has a constant speed throughout the journey, while Person 2's speed changes as a function of time.

(Determine who arrives at point Q first)

Since Person 1 has a constant speed of \(\frac{1}{2}\pi\), it will take \(\frac{4\pi}{\frac{1}{2}\pi} = 8\) units of time to reach point Q, given that the length of the path is \(4\pi\). For Person 2, we know the path is paraobolic. We can find the time it will take them to reach point Q by seeing when their x-coordinate becomes -4 (the x-coordinate of point Q): $$ 4 - t = -4 $$ Solving for t, we get \(t = 8\). So, both people reach point Q at the same time, at t = 8 units of time.

Key concepts

Graphing Paths

When graphing parametric paths, you are essentially visualizing how objects move in a plane over time. Consider the given paths in the exercise, which are defined by parametric equations. For the first path, the equation \( \mathbf{r}(t) = (4 \cos (\pi t / 8), 4 \sin (\pi t / 8)) \) represents a semicircular path. This is because the parametric equations use cosine and sine functions that naturally describe circular motions. As you vary \( t \) from 0 to 8, you would see half of a circle traced out, starting from \((4,0)\) and ending at \((-4,0)\).
\
For the second path, represented by \( \mathbf{R}(t) = (4-t, (4-t)^2 - 16) \), it traces a parabolic path in the plane. In both scenarios, using graphing tools such as Desmos or GeoGebra can significantly aid in visualizing these paths. By inputting these equations, you will see the semicircle and parabola visually connecting points \(P\) and \(Q\), helping you understand the motion between these points.

Velocity Vectors

Velocity vectors are crucial as they depict the direction and speed of an object traveling along a path. Essentially, the derivative of a parametric path equation will give you the velocity vector.
The velocity vector for the semicircular path \( \mathbf{r}(t) \) is derived as \( \mathbf{r}'(t) = (-\frac{1}{2}\pi\sin(\frac{\pi t}{8}), \frac{1}{2}\pi\cos(\frac{\pi t}{8})) \). This vector indicates how the object moves at each point along the circle.
Similarly, for the parabolic path \( \mathbf{R}(t) \), the velocity vector is \( \mathbf{R}'(t) = (-1, 2t - 8) \). This shows how quickly the position changes with respect to time for that specific path.
Understanding velocity vectors can help analyze motion characteristics, such as constant versus variable speed and the direction of movement at any moment.

Constant Speed

Constant speed implies that the magnitude of the velocity is unchanging over time. For the first person traveling along the semicircular path, their speed function is \( v_1(t) = |\mathbf{r}'(t)| = \frac{1}{2}\pi \). This constant value means that their rate of movement stays the same throughout their journey from \(P\) to \(Q\).
In contrast, the second person, traveling along the parabolic path, has a speed function \( v_2(t) = \sqrt{1 + (2t-8)^2} \). As observed, this speed varies as a function of time \(t\), meaning that their pace changes at each moment based on the derivative magnitude. The changing speed implies acceleration or deceleration might occur as they travel between \(P\) and \(Q\).

• A constant speed path provides a predictable travel time.
• Variable speeds can be challenging to account for when determining exact travel times without calculations.

Understanding the significance of these speeds assists in visualizing motion's steadiness or variability.

Parametric Curves Analysis

Parametric curves offer a versatile way to trace paths and movements within planes, allowing analysis beyond simple Cartesian coordinates. In exercises like the one in question, analyzing these curves combines mathematical function analysis with visual interpretation.
Begin with the understanding that each part of a parametric equation (such as \((x(t), y(t))\)) serves a role in tracing a distinct path. For the semicircular path, the coefficients and functions create smooth arcs due to trigonometric relationships. Trigonometric parametrics are ideal for circular paths.
The parabolic path emphasizes polynomial properties, where quadratic variations provide a different type of curvature. Knowing this, parametric curves illuminate diverse path shapes: circles, ellipses, hyperbolas, and more.
Analyzing these curves allows you to provide answers to questions like distance traveled and time to complete a path, emphasizing the location shifts at specific times.

• Parametrics expand visualization and problem-solving possibilities.
• They play critical roles in physics, engineering, and graphics for modeling real-world scenarios.

Developing a clear understanding of parametric analysis can greatly enhance one's ability to predict and calculate path trajectories.