Problem 41 - Essential Calculus Early Transcendentals

Question

A particle is moving with the given data. Find the position of the particle. $$a(t)=10 \sin t+3 \cos t, \quad s(0)=0, \quad s(2 \pi)=12$$

Step-by-Step Solution

Answer

The position of the particle is given by \(s(t) = -10 \sin t - 3 \cos t + \frac{6}{\pi} t + 3 \).

Step 1: Establish the Relationship

Given the acceleration function \(a(t) = 10 \sin t + 3 \cos t\), we know that acceleration is the second derivative of displacement (position). Thus, to find the position function, we first need to integrate the acceleration function to find the velocity function.

Step 2: Find the Velocity Function

Integrate the acceleration function \(a(t) = 10 \sin t + 3 \cos t\). \[ v(t) = \int (10 \sin t + 3 \cos t) \, dt = -10 \cos t + 3 \sin t + C \]Here, \(C\) is the constant of integration.

Step 3: Integrate to Find Position Function

Integrate the velocity function to find the position function:\[ s(t) = \int (-10 \cos t + 3 \sin t + C) \, dt \]\[ s(t) = -10 \sin t - 3 \cos t + Ct + D \]Where \(D\) is another constant of integration.

Step 4: Apply Initial Conditions

Use the initial conditions to solve for the constants \(C\) and \(D\). Given \(s(0) = 0\):\[ 0 = -10 \sin(0) - 3 \cos(0) + C(0) + D \]\[ 0 = -3 + D \]\[ D = 3 \]

Step 5: Use Final Condition

Apply the condition \(s(2\pi) = 12\) to find \(C\).\[ 12 = -10 \sin(2\pi) - 3 \cos(2\pi) + C(2\pi) + 3 \]\[ 12 = -3 + 2\pi C + 3 \]\[ 12 = 2\pi C \]\[ C = \frac{12}{2\pi} = \frac{6}{\pi} \]

Step 6: Write Down the Position Function

Having found \(C\) and \(D\), substitute them back into the expression for \(s(t)\).\[ s(t) = -10 \sin t - 3 \cos t + \frac{6}{\pi} t + 3 \]

Key Concepts

Acceleration Function

An acceleration function, like the one given as \(a(t) = 10 \sin t + 3 \cos t\), describes how the velocity of an object changes over time. The acceleration function is crucial in determining how a particle moves along a path since it’s directly related to the force acting on it.
To fully understand the particle's motion, it's essential to memorize that:

• The acceleration function is the second derivative of the position function \(s(t)\).
• It describes the rate of change of velocity, meaning it tells us how quickly the velocity is changing at any given point.

To get to the position function from acceleration, we need to solve the inverse problem: starting from acceleration to find velocity and ultimately, position by using integration.

Integration

Integration is the mathematical process we use to work backward from acceleration to velocity and from velocity to position. It is essentially finding the antiderivative of a function. Here, we have two integrals to work with:

• First, integrate the acceleration function to find the velocity function: \ \( v(t) = \int (10 \sin t + 3 \cos t) \, dt = -10 \cos t + 3 \sin t + C \)
• Next, integrate this velocity function to get the position function: \ \( s(t) = \int (-10 \cos t + 3 \sin t + C) \, dt = -10 \sin t - 3 \cos t + Ct + D \)

Integration introduces constants \(C\) and \(D\) which are necessary to consider the boundaries or initial conditions of the problem. This process is vital because it gradually simplifies the information obtained from acceleration to a tangible position function that tells us where the object is at any given time.

Initial Conditions

Initial conditions are specific values of the function given for certain points in time. They are critical because they allow us to solve for unknown constants introduced during integration. In our example, we used initial conditions to find constants \(C\) and \(D\).

• Given \(s(0) = 0\), substitute into the position function to solve for \(D\). This eliminates variables and simplifies to \(D = 3\).
• Another condition, \(s(2\pi) = 12\), is used to find \(C\). Solving it yields \(C = \frac{6}{\pi}\).

By applying these initial and final conditions, we tailor the general solution to fit the specific motion described, providing the exact position function that satisfies both given scenarios.

Position Function

Finally, the position function \(s(t)\) describes the exact position of the particle at any time \(t\). With the constants solved via initial conditions, we can write the complete function:

• \( s(t) = -10 \sin t - 3 \cos t + \frac{6}{\pi} t + 3 \)

This equation tells us the particle's location at any time, based on its initial acceleration. It fully incorporates the trigonometric components resulting from acceleration and the linear component derived during integration.
Remember, the position function is a pivotal result of integrating the acceleration and velocity functions. It provides insight into the path and behavior of the moving particle throughout its journey.