Problem 44 - Thomas Calculus in SI Units

Question

Give the velocity \(v=d s / d t\) and initial position of an object moving along a coordinate line. Find the object's position at time \(t\). $$v=\frac{2}{\pi} \cos \frac{2 t}{\pi}, \quad s\left(\pi^{2}\right)=1$$

Step-by-Step Solution

Answer

The position function is \( s(t) = \sin \frac{2t}{\pi} + 1 \).

Step 1: Identify the given information

We are given the velocity function of an object moving along a coordinate line: \( v = \frac{2}{\pi} \cos \frac{2t}{\pi} \). Additionally, we know the initial position at a specific time: \( s(\pi^2) = 1 \). Our goal is to find the position function \( s(t) \) of the object at any time \( t \).

Step 2: Set up the integral to find the position function

To find the position function \( s(t) \), we need to integrate the velocity function with respect to \( t \). The velocity function \( v = \frac{2}{\pi} \cos \frac{2t}{\pi} \) can be integrated to find the displacement. Thus, we write the integral:\[s(t) = \int \frac{2}{\pi} \cos \frac{2t}{\pi} \, dt + C.\]

Step 3: Perform the integration

We perform the integration by making use of the known integral of the cosine function. The integral of \( \cos(kt) \) is \( \frac{1}{k} \sin(kt) \). Let \( k = \frac{2}{\pi} \), then:\[s(t) = \left[ \sin \frac{2t}{\pi} \right] + C.\]

Step 4: Include the constant of integration

The integral gives us a general solution including the constant \( C \):\[s(t) = \sin \frac{2t}{\pi} + C.\]

Step 5: Use the initial condition to determine the constant

We know that at \( t = \pi^2 \), the position \( s(\pi^2) = 1 \). Substitute this into the equation to find \( C \):\[1 = \sin \frac{2\pi^2}{\pi} + C \Rightarrow 1 = \sin(2\pi) + C.\]Since \( \sin(2\pi) = 0 \), it simplifies to \( C = 1 \).

Step 6: Write the final position function

Substitute the constant \( C \) back into the position function to get the final expression for \( s(t) \):\[s(t) = \sin \frac{2t}{\pi} + 1.\]

Key Concepts

Velocity and position

When an object moves along a coordinate line, its velocity describes how its position changes over time. Velocity is essentially the rate of change of the object's position with respect to time and is given by the derivative of the position function, or \( v(t) = \frac{ds}{dt} \).
To find the position function from the velocity, you need to integrate the velocity function with respect to time.
The position gives us the specific location of the object at any given time. In this exercise, you were given the velocity function \( v(t) = \frac{2}{\pi} \cos \frac{2t}{\pi} \) and the initial position \( s(\pi^2) = 1 \).
From this, the aim is to determine the expression for \( s(t) \) at any future time \( t \).

• Velocity function describes how fast and in which direction position changes.
• Integrating velocity gives the change in position, known as displacement.
• Initial position is crucial to solve for the constant of integration.

Integration

Integration is the key process used to find the position function from a velocity function. It helps us "accumulate" all the tiny changes in position given by the velocity over time.
The integral of the velocity function \( v = \frac{2}{\pi} \cos \frac{2t}{\pi} \) results in the position function\( s(t) \).
When we integrate, we often obtain an extra term, a constant, \( C \), since the indefinite integral represents a family of functions.This constant \( C \) is determined using initial conditions, like the given initial position \( s(\pi^2) = 1 \).
The steps to find the integration are:

• Identify the given velocity function.
• Write the integral expression to find displacement.
• Perform the integration using known integral formulas.
• Apply initial conditions to find the integration constant.
• Construct the final position function.

Integration is not just mathematical manipulation, it represents finding the area under the curve of the velocity function, which equates to the total change in position.

Trigonometric functions

Trigonometric functions like sine and cosine are prevalent in many calculus problems, especially those involving periodic or oscillating systems. Here, the velocity function uses the cosine function, specifically \( \cos \frac{2t}{\pi} \), which is characteristic of oscillating systems.
In the integration process, knowing the integrals of trigonometric functions is essential. For instance, the integral of \( \cos(kx) \) is \( \frac{1}{k} \sin(kx) \), where \( k \) is the inside constant for the trigonometric argument.

• Cosine and sine functions model oscillation and wave phenomena.
• The integration shifts a cosine model into a sine model over time.
• Using properties of trigonometric functions simplifies manipulation during calculus operations.

Understanding these functions allows us to describe and predict system behaviors over time effectively. This makes trigonometric calculus an essential tool in physics and engineering fields.