Problem 44 - Thomas Calculus

Question

Motion on a line The positions of two particles on the \(s\) -axis are \(s_{1}=\sin t\) and \(s_{2}=\sin (t+\pi / 3),\) with \(s_{1}\) and \(s_{2}\) in meters and \(t\) in seconds. \begin{equation} \begin{array}{l}{\text { a. At what time(s) in the interval } 0 \leq t \leq 2 \pi \text { do the particles }} \\ \quad {\text { meet? }} \\ {\text { b. What is the farthest apart that the particles ever get? }} \\ {\text { c. When in the interval } 0 \leq t \leq 2 \pi \text { is the distance between the }} \\ \quad {\text { particles changing the fastest? }}\end{array} \end{equation}

Step-by-Step Solution

Answer

a. Particles meet at \( t = \frac{\pi}{2} \). b. Farthest apart when \( |\cos(t + \frac{\pi}{6})| = 1 \). c. Fastest distance change at \( t = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \).

Step 1: Find time(s) when particles meet

To find when the particles meet, set their positions equal: \( \sin t = \sin(t + \frac{\pi}{3}) \). Use the sine difference identity: \( \sin A - \sin B = 2 \cos\left( \frac{A+B}{2} \right)\sin\left( \frac{A-B}{2} \right) = 0\). This implies either \( \cos\left( \frac{2t + \frac{\pi}{3}}{2} \right) = 0 \) or \( \sin\left( -\frac{\pi}{6} \right) = 0 \). The solutions for \( t \) in \( 0 \leq t \leq 2\pi \) are \( t = \frac{n\pi}{3} - \frac{\pi}{6} \) with valid integers \( n \), giving \( t= \frac{\pi}{2}, \frac{5\pi}{2} \). However, only \( t= \frac{\pi}{2} \) is in the range.

Step 2: Determine maximum distance between particles

Calculate the distance function \( D(t) = |\sin t - \sin(t + \frac{\pi}{3})| \).Simplify using the identity for sine difference: \( D(t) = 2|\cos\left(t + \frac{\pi}{6}\right) \sin\left(\frac{\pi}{6}\right)| = |\cos\left(t + \frac{\pi}{6}\right)| \).The maximum value of \( |\cos\theta| \) is 1, so look for \( t \) such that \( t + \frac{\pi}{6} = n\pi \). This occurs when \( t = n\pi - \frac{\pi}{6} \), with valid \( t \) values in the interval such as \( t = \frac{\pi}{6}, \frac{7\pi}{6} \).

Step 3: Find when distance changes fastest

Differentiate the distance function to see when it is changing fastest. \( D'(t) = -\sin t \cos(t + \frac{\pi}{3}) - \cos t \sin(t + \frac{\pi}{3}) = -\sin(t + \frac{\pi}{3} - t) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} \). The fastest change corresponds to the maximum of the absolute value of \( D'(t) \), which hints at periodic points in the sine wave. Within given bounds, check multiples of \( \pi \), determining extremes using basic sine behavior: \( t = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6} \).

Key Concepts

Position on an axis

When dealing with particle motion, it's essential to understand what it means for a particle to have a position on an axis. In this exercise, the position is given by functions of time, specifying the location of particles on the s-axis.

• The position of the first particle is given by \(s_1 = \sin t\),
• while the second particle's position is specified by \(s_2 = \sin(t + \frac{\pi}{3})\).

Here, these functions are trigonometric in nature due to the sine terms. The axis represents a linear path, and the positions correspond to points on this path, moving according to the value of time \(t\). Recognizing position as a time-dependent function is foundational, allowing us to find where particles meet and how far apart they are at various times.

Trigonometric identities

Trigonometric identities play a crucial role in simplifying and solving equations involving trigonometric functions. For this exercise, knowing how to use these identities becomes instrumental in finding when and where our particles meet or are farthest apart. One critical identity used here is the sine difference identity:\[\sin A - \sin B = 2 \cos\left( \frac{A+B}{2} \right)\sin\left( \frac{A-B}{2} \right).\]This transforms the equation \( \sin t = \sin(t + \frac{\pi}{3}) \) into a form that helps isolate values of \(t\). Such identities allow us to handle expressions that appear complex initially, turning them into something more manageable through conversion into products and trigonometric expansions. Understanding these identities is like having a toolkit to solve various trigonometric challenges in mathematics.

Distance calculation

Calculating the distance between two moving particles involves taking the absolute difference of their position functions. The expression for distance helps to determine how far two particles are from each other at any point in time.In this case, the distance function is:\[D(t) = |s_1 - s_2| = |\sin t - \sin(t + \frac{\pi}{3})|.\]By using trigonometric identities, like the sine difference identity, you simplify this to:\[D(t) = 2|\cos(t + \frac{\pi}{6}) \sin(\frac{\pi}{6})| = |\cos(t + \frac{\pi}{6})|.\]Recognizing the pattern that \(|\cos \theta|\) achieves its maximum value of 1 is pivotal, allowing us to conclude the maximum distance without needing a laborious computation for every point in time. This simplification aids in quickly assessing when two moving points reach their extremes in terms of proximity.

Differentiation

Differentiation is a fundamental calculus tool used to find the rate of change of a function. In particle motion, it helps determine when a function's behavior changes fastest, which, in this case, relates to the rate at which the distance between two particles varies.To identify when the distance between particles changes fastest, the first derivative of the distance function is required:\[D'(t) = -\sin t \cos(t + \frac{\pi}{3}) - \cos t \sin(t + \frac{\pi}{3}).\]This derivative indicates the rate of change of the distance over time. By setting up the derivative and analyzing parts of the periodic sine and cosine waves, you determine intervals where this rate of change is maximized. Understanding the application of differentiation in this context helps to predict behavior efficiently without extensive computation.

Periodic functions

Periodic functions are functions that repeat their values in regular intervals or periods. Both sine and cosine function as classic examples of periodic functions encountered often in physics and mathematics.In our problem, the positions of the particles are described by sine functions, which inherently have a period of \(2\pi\). As such, knowing their periodic nature is extremely helpful because:

• It provides insight into the behavior and movements of particles over predictable intervals.
• Helps determine potential meeting points and where and when distances are maximized or changing fastest.

By understanding their periodicity, we can forecast how the sine function will behave beyond the initial plotted interval. This is crucial for both graphical interpretations and when generalizing solutions to continue observing conditions even after their basic cycle is complete.