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. (a) At what time \((\mathrm{s})\) in the interval \(0 \leq t \leq 2 \pi\) do the particles meet? (b) What is the farthest apart that the particles ever get? (c) When in the interval \(0 \leq t \leq 2 \pi\) is the distance between the particles changing the fastest?

Short answer

The particles intersect at \(t=0\), \(t=\frac{4\pi}{3}\) and \(t=2\pi\). The farthest distance occurs at \(t=\frac{2\pi}{3}\) and \(t=\frac{5\pi}{3}\), with a distance of \(\frac{\sqrt{3}}{2}\). The distance between the particles changes the fastest at \(t=\frac{\pi}{3}\), \(t=\pi\), and \(t=\frac{5\pi}{3}\).

Step-by-step solution

Determine When the Particles Meet

The two particles meet when their positions are equal, that is when \(s_1 = s_2\). Set \(\sin(t) = \sin(t+\frac{\pi}{3})\) and solve for \(t\).

Calculate the Farthest Distance Between the Particles

To find the farthest distance between the particles, we take the difference of the two sinusoidal functions to have a new function \(s = s_1 - s_2 = \sin(t) - \sin(t+\frac{\pi}{3})\). Then, we find the maximum of the absolute value of this function over the given interval.

Find When Distance Changes Fastest

The rate at which the distance between particles changes is found by calculating the derivative of the distance function obtained in the previous step. So, take the derivative of \(s\) denoted by \(s'\), and find its maximum over the defined interval.

Key concepts

Sinusoidal Functions in Motion

Sinusoidal functions, such as sine and cosine, are foundational in describing periodic motion – the kind that repeats in a predictable cycle. These functions model a vast array of phenomena, including sound waves, alternating current, and in our particular case, the motion of particles along a line.

In the given exercise, the position of two particles is described by sinusoidal functions, which vary with time, making them perfect to represent motion on a line. The sine function, denoted as \(\sin(t)\), oscillates between -1 and 1, providing a smooth wave motion. When a phase shift, like \(\frac{\pi}{3}\), is introduced, it causes one particle to be out of step with the other, creating a lag in the motion. This shift is essential to determining when and how the particles meet or diverge over time.

To visualize sinusoidal motion, imagine a point moving around a circle at a constant rate. Projecting this point's vertical position onto a line as the circle rotates results in the sine wave pattern. In the context of our problem, this could resemble the back-and-forth motion of a pendulum or the oscillating path of a particle in a spring-mass system.

Derivative to Find Changing Rates

The power of calculus shines when we want to understand how things change over time, a key point in many scientific and engineering contexts. The derivative is a mathematical tool that takes a function and provides another function that tells us the rate of change at any point.

In the context of particle motion, if the position of a particle is given by a function of time \(s(t)\), then the particle's velocity - how fast and in what direction it's moving - is given by the derivative of that function, denoted as \(s'(t)\). Similarly, by taking the derivative of the velocity function, we could find the acceleration, detailing how the velocity is changing over time.

In our exercise, to find when the distance between the particles changes the fastest, we calculate the derivative of the distance function. Intuitively, this will tell us the rate at which the particles are moving apart or together. The peak rate of change, a maximum or minimum of the derivative, indicates the points in time when the distance between particles is changing most rapidly.

Maxima and Minima of Functions

Identifying the highest or lowest points on a graph, known as the maxima and minima, respectively, is a crucial application of derivatives in calculus. The points of maximum and minimum values provide key insights into the behavior of functions and, by extension, the physical systems they represent.

To find these points, one takes the derivative of the function and sets it equal to zero. This gives us potential candidates known as critical points. However, not all critical points are maxima or minima, which is why further testing with second derivatives or comparing values is often necessary.

In our particle motion problem, we are interested in two things: when the particles are farthest apart and when their rate of distance change is the fastest. For the former, we seek the maxima of the absolute value of the difference in their positions. For the latter, we look for the maximum value of the derivative of this distance function. These calculations allow us to predict the most significant moments of separation and convergence over their periodic journey.