Problem 45 - University Calculus: Early Transcendentals

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 \(t\). $$v=\sin \pi t, \quad s(0)=0$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are given the velocity function \(v = \sin(\pi t)\) and the initial position \(s(0) = 0\). We need to find the object's position function \(s(t)\) at any time \(t\). Velocity is the derivative of position, so we're essentially finding the antiderivative or integral of velocity to obtain the position.

Step 2: Integrate the Velocity Function

To find \(s(t)\), we integrate the velocity function \(v(t)\) with respect to time \(t\). This means we need to compute the integral:\[s(t) = \int \sin(\pi t) \, dt.\]

Step 3: Calculate the Integral

The integral of \(\sin(\pi t)\) with respect to \(t\) is calculated as follows: \[s(t) = -\frac{1}{\pi}\cos(\pi t) + C,\] where \(C\) is the constant of integration. This result follows from the integration rule for \(\sin(kt)\) which is \(-\frac{1}{k}\cos(kt)\).

Step 4: Apply Initial Condition

We use the initial condition \(s(0) = 0\) to find \(C\). Substitute \(t = 0\) into the integral result:\[-\frac{1}{\pi}\cos(\pi \cdot 0) + C = 0.\] Since \(\cos(0) = 1\), the equation simplifies to \[-\frac{1}{\pi} + C = 0.\] Solve for \(C\) to find that \(C = \frac{1}{\pi}\).

Step 5: Write the Final Position Function

Substituting \(C = \frac{1}{\pi}\) back into the position function gives us:\[s(t) = -\frac{1}{\pi}\cos(\pi t) + \frac{1}{\pi}.\] This is the position function of the object at time \(t\).

Key Concepts

Velocity Function

The velocity function defines the rate of change of an object's position over time. It tells us how fast an object is moving and in which direction. In the problem, the velocity function given is \(v = \sin(\pi t)\). This means the object's speed and direction change following a sine wave. The velocity function relates directly to the derivative of the position function.
Understanding this function is crucial, since the primary goal of the exercise is to use it to find the function describing the object's position at any time \(t\). When a problem provides a velocity as a function of time, you're actually given the derivative of the position function. This is key for finding the position.

Antiderivative

The antiderivative is essentially the process we use to find the original function from its derivative. This process is also referred to as integration. In this problem, the velocity function \(v = \sin(\pi t)\) is the derivative of the position function we are searching for, noted as \(s(t)\).
To find the position function, we integrate the given velocity function. Integrating \(\sin(\pi t)\) with respect to \(t\), we get \(-\frac{1}{\pi}\cos(\pi t) + C\), where \(C\) is the constant of integration. This constant arises because the process of differentiation loses the constant term, which must be determined separately.

Initial Condition

Once we've found the antiderivative, we must find the constant of integration \(C\). This is done using the initial condition provided in the problem, often expressed as \(s(0) = 0\).
By substituting \(t = 0\) into the integrated function \(-\frac{1}{\pi}\cos(\pi t) + C\) and setting it equal to the initial condition, \(0\), we can solve for \(C\).
This gives \(-\frac{1}{\pi} \times 1 + C = 0\), which simplifies to \(C = \frac{1}{\pi}\). Initial conditions are crucial because they tie the abstract, indefinite antiderivative back to the specific problem, allowing us to finalize the position function.

Position Function

The position function \(s(t)\) indicates an object's location at any time \(t\). After integrating the velocity function and applying the initial condition, we find the complete position function.
For this problem, substituting the constant \(C = \frac{1}{\pi}\) back into the integrated function gives us the final position function:

• \(s(t) = -\frac{1}{\pi}\cos(\pi t) + \frac{1}{\pi}\)

This equation now provides a direct connection between time and the position of the moving object. The position function captures both the pattern of movement described by the velocity and respects the initial position condition, fully solving the problem.