Problem 39 - Essential Calculus Early Transcendentals

Question

A particle is moving with the given data. Find the position of the particle. $$v(t)=\sin t-\cos t, \quad s(0)=0$$

Step-by-Step Solution

Answer

The position of the particle is \( s(t) = -\cos t - \sin t + 1 \).

Step 1: Understand the Problem

We are given the velocity function of a particle, \( v(t) = \sin t - \cos t \), and an initial position \( s(0) = 0 \). The task is to find the position function \( s(t) \) by integrating the velocity function.

Step 2: Set Up the Integral Equation

To find the position function \( s(t) \), we need to integrate the velocity function \( v(t) \). This can be expressed as \( s(t) = \int v(t) \, dt = \int (\sin t - \cos t) \, dt \).

Step 3: Integrate the Velocity Function

We need to integrate each term separately. The integral of \( \sin t \) is \( -\cos t \), and the integral of \( -\cos t \) is \( -\sin t \). Therefore, the indefinite integral is \( s(t) = -\cos t - \sin t + C \), where \( C \) is the constant of integration.

Step 4: Apply the Initial Condition

Use the initial condition \( s(0) = 0 \) to find the constant \( C \). Substitute \( t = 0 \) in the equation: \( 0 = -\cos(0) - \sin(0) + C \). Since \( \cos(0) = 1 \) and \( \sin(0) = 0 \), we have \( 0 = -1 + C \). Thus, \( C = 1 \).

Step 5: Write the Position Function

With the value of \( C \) determined, the position function is \( s(t) = -\cos t - \sin t + 1 \). This represents the position of the particle for any time \( t \).

Key Concepts

Velocity Function

The velocity function describes the rate of change of a particle's position with respect to time. In this exercise, we are given the velocity function,

• \( v(t) = \sin t - \cos t \)

This equation helps us understand how the speed and direction of the particle change at any given time \( t \). By knowing the velocity, we can determine how far the particle has moved from its starting point over a certain period.
Think of velocity as the speedometer in a car. It tells you how fast you're going at any moment.
This function is the derivative of the position function, meaning it shows how position changes over time.

Integration

Integration is a fundamental concept in calculus used to find accumulated quantities. In this context, integrating the velocity function retrieves the position function.

• To find the position function, compute the integral of \( v(t) \) with respect to \( t \).

Mathematically, it is expressed as,

• \( s(t) = \int v(t) \, dt = \int (\sin t - \cos t) \, dt \)

Each term of the velocity function is integrated separately:

• The integral of \( \sin t \) is \( -\cos t \).
• The integral of \( -\cos t \) is \( -\sin t \).

Don't forget the constant of integration \( C \), which appears because velocity only gives the rate of change, not the exact position.
Integration is like adding up all tiny changes along the way to find the total change in the particle's position.

Position Function

The position function gives the exact position of a particle at any time \( t \). Once the velocity is integrated, you get an equation that models the particle's position.

• \( s(t) = -\cos t - \sin t + C \)

This formula demonstrates the result of integrating the velocity function \( v(t) = \sin t - \cos t \).
The position function reflects where the particle is located over time based on its starting point and speed changes.
It's like a GPS showing the particle's location on a map at any moment.

Initial Condition

An initial condition in the context of differential equations provides a specific value for \( s(t) \) at a particular point, often where \( t = 0 \).

• For this problem, the initial condition is \( s(0) = 0 \).

This value helps determine the constant of integration \( C \), ensuring the position function accurately represents the particle's path.

• The equation \( 0 = -1 + C \) simplifies to \( C = 1 \), given \( \cos 0 = 1 \) and \( \sin 0 = 0 \).

With \( C \) found, the final position function becomes:

• \( s(t) = -\cos t - \sin t + 1 \).

The initial condition is like setting the starting point for a runner on a track. It tells you exactly where the journey began.