Problem 79 - Calculus Single Variable

Question

The position of an oscillating body is given by \(p(t)=\) \(\sin (2 t+\pi / 6) .\) Calculate the average velocity of the body over a time interval of the form \([0, \Delta t]\) for \(\Delta t=10^{-n}\) \(n=0,1,2,3\) and \(4 .\) Display your results in the form of a table. Formulate a guess \(v\) for the instantaneous velocity of the body at time \(t=0 .\) Plot \(p .\) In the same coordinate plane, add the graph of the straight line that passes through \((0,1 / 2)\) and that has slope \(v\). Does the resulting figure support your conjectured value? Explain.

Step-by-Step Solution

Answer

The table indicates the trend in average velocity. The conjectured velocity, based on curve analysis, approaches 2 at \(t = 0\). The slope of 2 fits as \(p'(0)\), suggesting the line adequately approximates \(p(t)\) at \(t = 0\).

Step 1: Calculate Initial Position for t = 0

To find the initial position of the body at time \(t = 0\), substitute \(t = 0\) in the position function: \[ p(0) = \sin\left(2 \cdot 0 + \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}. \]

Step 2: Define the Average Velocity Formula

The average velocity \(v_{avg}\) over the interval \([0, \Delta t]\) is given by the change in position over the change in time: \[ v_{avg} = \frac{p(\Delta t) - p(0)}{\Delta t}. \]

Step 3: Compute Average Velocity for Δt = 1

Set \(\Delta t = 10^{-0} = 1\). Calculate the average velocity: Substitute \(\Delta t = 1\) into the position function: \[ p(1) = \sin\left(2 \cdot 1 + \frac{\pi}{6}\right) = \sin\left(2 + \frac{\pi}{6}\right), \] and \[ v_{avg} = \frac{\sin\left(2 + \frac{\pi}{6}\right) - \frac{1}{2}}{1}. \] Compute the value to fill the table.

Step 4: Compute Average Velocity for Δt = 0.1

Set \(\Delta t = 10^{-1} = 0.1\). Calculate the average velocity: \[ p(0.1) = \sin\left(2 \cdot 0.1 + \frac{\pi}{6}\right) = \sin\left(0.2 + \frac{\pi}{6}\right), \] and \[ v_{avg} = \frac{\sin\left(0.2 + \frac{\pi}{6}\right) - \frac{1}{2}}{0.1}. \] Compute the value to fill the table.

Step 5: Compute Average Velocity for Δt = 0.01

Set \(\Delta t = 10^{-2} = 0.01\). Calculate the average velocity: \[ p(0.01) = \sin\left(2 \cdot 0.01 + \frac{\pi}{6}\right) = \sin\left(0.02 + \frac{\pi}{6}\right), \] and \[ v_{avg} = \frac{\sin\left(0.02 + \frac{\pi}{6}\right) - \frac{1}{2}}{0.01}. \] Compute the value to fill the table.

Step 6: Compute Average Velocity for Δt = 0.001

Set \(\Delta t = 10^{-3} = 0.001\). Calculate the average velocity: \[ p(0.001) = \sin\left(2 \cdot 0.001 + \frac{\pi}{6}\right) = \sin\left(0.002 + \frac{\pi}{6}\right), \] and \[ v_{avg} = \frac{\sin\left(0.002 + \frac{\pi}{6}\right) - \frac{1}{2}}{0.001}. \] Compute the value to fill the table.

Step 7: Compute Average Velocity for Δt = 0.0001

Set \(\Delta t = 10^{-4} = 0.0001\). Calculate the average velocity: \[ p(0.0001) = \sin\left(2 \cdot 0.0001 + \frac{\pi}{6}\right) = \sin\left(0.0002 + \frac{\pi}{6}\right), \] and \[ v_{avg} = \frac{\sin\left(0.0002 + \frac{\pi}{6}\right) - \frac{1}{2}}{0.0001}. \] Compute the value to fill the table.

Step 8: Construct the Table of Average Velocities

Summarize the results from Steps 3-7 in a table with two columns: \(\Delta t\) and the calculated \(v_{avg}\).

Step 9: Guessed Instantaneous Velocity

By examining the trend of the average velocities as \(\Delta t\) decreases, conjecture the instantaneous velocity \(v\) at \(t = 0\) based on how \(v_{avg}\) approaches as \(\Delta t \to 0\).

Step 10: Plot the Functions

Plot the function \(p(t) = \sin(2t + \frac{\pi}{6})\). Also plot the line passing through \((0, \frac{1}{2})\) with the slope equal to the guessed instantaneous velocity \(v\).

Step 11: Analyze the Plot

Determine whether the line approximates the tangent to the curve at \(t = 0\). If the line closely matches the curve at \(t = 0\), it supports the conjectured value for \(v\).

Key Concepts

Oscillating Motion

Oscillating motion describes a repetitive back-and-forth movement around an equilibrium point. Imagine a swinging pendulum or a vibration of a guitar string. These are classic examples of oscillating systems. In our exercise, the position of the oscillating body is defined by the function \(p(t) = \sin(2t + \frac{\pi}{6})\).
Oscillations typically follow a regular pattern or cycle. The sine function captures this nature effectively, allowing us to model movement periods. Such motion can be influenced by parameters like amplitude and frequency, determining characteristics like how far and how fast the body oscillates.
By studying these models, we learn how oscillatory behaviors occur not only in classroom scenarios but in real-world physics, engineering, and even nature. Recognizing these patterns can provide insights into predicting and controlling various types of mechanical actions.

Instantaneous Velocity

Instantaneous velocity tells us how fast an object is moving at any specific point in time, different from average velocity that considers wider time spans. It's akin to what a speedometer reads at any instant while driving a car.
In the context of our exercise, the challenge is to determine this velocity at time \(t = 0\) for the oscillating body. As we calculate the average velocities for increasingly smaller durations (\(\Delta t\) approaching zero), we begin to approach the instantaneous velocity.
Using calculus, we can take the derivative of the position function to find the velocity function. This velocity function, evaluated at \(t = 0\), provides the required instantaneous velocity. In this case, you would take the derivative of the sine function defined in the exercise to gain insight into the immediate speed of change at the starting point.

Sine Function

The sine function, \(\sin(x)\), is a fundamental part of trigonometry that displays a wave-like pattern. It oscillates between -1 and 1, crossing the x-axis at intervals of \(\pi\) radians, making it ideal for modeling periodic and oscillatory motion.
In the context of this exercise, \(p(t)=\sin(2t+\frac{\pi}{6})\) acts as the position function of the oscillating body. Here, the coefficient 2 in \(2t\) signifies that the oscillation rate is doubled compared to the base sine function. Meanwhile, \(\frac{\pi}{6}\) works as a phase shift, indicating that the wave moves to the left on a graph.
Understanding how the sine function manipulates time-based attributes of oscillation helps students appreciate broader applications in physics and engineering, as well as how we apply these concepts in technology like signal processing.

Calculus

Calculus is a branch of mathematics focusing on the rates of change and accumulation. It's essential for understanding dynamic systems, such as the motion we've been examining in this exercise.
Here, calculus aids us in determining both the average and instantaneous velocities of an oscillating body. By taking derivatives, we essentially "uncover" the rate at which position changes over time (speed or velocity).
In our particular example, deriving the position function \(p(t)\) gives us a velocity function. The derivative helps pinpoint not just the average movement across time intervals, but the precise rate at a specific instant, illuminating the intricate dance of variables over the course of a unit of time.
Mastering calculus involves understanding its basic principles, such as derivatives and integrals, and how they apply to real-world scenarios, ultimately providing powerful tools for analysis in science, engineering, and beyond.