Problem 44 - 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 \(t\). $$v=32 t-2, \quad s(0.5)=4$$

Step-by-Step Solution

Answer

The position function is \(s(t) = 16t^2 - 2t + 1\).

Step 1: Understand the Given Formula for Velocity

The velocity of an object is given by the formula \(v = \frac{ds}{dt}\). In this case, the velocity is \(v = 32t - 2\). This means the velocity is a function of time \(t\). Our goal is to find the object's position \(s(t)\) as a function of time.

Step 2: Set Up the Integral to Find Position

To find the position \(s(t)\), we need to integrate the velocity function with respect to time \(t\). The formula for the position based on the velocity is: \[s(t) = \int v(t) \, dt = \int (32t - 2) \, dt\]This integral will give us the general position function.

Step 3: Perform the Integration

Integrate the velocity function: \[\int (32t - 2) \, dt = 32 \int t \, dt - \int 2 \, dt\]. The integral of \(32t\) with respect to \(t\) is \(16t^2\), and the integral of \(-2\) is \(-2t\). Thus, the general position function is:\[s(t) = 16t^2 - 2t + C\]where \(C\) is the constant of integration.

Step 4: Use the Initial Condition to Solve for C

We are given that \(s(0.5) = 4\). Substitute \(t = 0.5\) and \(s = 4\) into the position function to solve for \(C\):\[4 = 16(0.5)^2 - 2(0.5) + C\]Calculate:\[4 = 16(0.25) - 1 + C\]\[4 = 4 - 1 + C\]\[4 = 3 + C\]Therefore, \(C = 1\).

Step 5: Write the Final Position Function

Substitute the value of \(C = 1\) back into the general formula for position:\[s(t) = 16t^2 - 2t + 1\].This is the function that describes the position of the object as a function of time.

Key Concepts

Velocity Function

The velocity function describes the rate of change of an object's position with respect to time. In simple terms, it tells us how fast the object is moving at any given moment. For this particular exercise, the velocity function is given as \(v = 32t - 2\). This means:

• The velocity changes linearly with time, \(t\).
• The object's speed increases as time progresses, at a rate of \(32\text{ units per time unit}\).
• The constant \(-2\) implies that initially (at \(t = 0\)) the object has a speed slower by \(2\) units.

Understanding velocity is crucial since it's a stepping stone to finding the position of the object.

Integration

Integration is a mathematical process that can be understood as the inverse operation of differentiation. It helps us find a function when we know its rate of change. In the context of this problem, we integrate the velocity function to get the position function. Here's how:

• The given velocity function \(v(t) = 32t - 2\) tells us the rate of change of position over time.
• To find the position, \(s(t)\), we integrate \(v(t)\) with respect to \(t\): \[s(t) = \int (32t - 2) \, dt\]
• After integration, we get: \[s(t) = 16t^2 - 2t + C\]

The result of the integration is the general form of the position function, which includes a constant \(C\) known as the constant of integration.

Initial Condition

An initial condition is a piece of information that allows us to solve for any unknown constants after performing integration. It's given so that we can find the exact position function that fits the specific scenario described in the problem. Here, the initial condition is \(s(0.5) = 4\).To use this:

• Substitute \(t = 0.5\) and \(s = 4\) into the integrated position formula: \[4 = 16(0.5)^2 - 2(0.5) + C\]
• Calculate and solve for \(C\): \[4 = 4 - 1 + C\], which simplifies to \(C = 1\).

This process ensures that our position function is accurate and reflects the initial conditions provided.

Constant of Integration

The constant of integration, often denoted as \(C\), arises when you integrate a function. This constant represents all possible vertical shifts of a function after integration, since derivatives of constants are zero.Here's how it works in the problem:

• After integrating the velocity function \(v(t) = 32t - 2\), we get the general form of the position equation \(s(t) = 16t^2 - 2t + C\).
• We use the initial condition \(s(0.5) = 4\) to find \(C\).
• Solving the initial condition gives \(C = 1\), specifying the position function's particular shift needed to match the given conditions.

Finding \(C\) is essential to accurately model the exact scenario described, ensuring our mathematical representation reflects the physical reality of the problem.