Question

Given the velocity of an object and its initial position, how do you find the position of the object, for \(t \geq 0 ?\)

Short answer

Question: Given the velocity function \(v(t)\) of an object and its initial position \(x_0\), determine the position function \(x(t)\) for \(t \geq 0\). Answer: The position function can be found by following these steps: 1. Integrate the velocity function: \(x(t) = \int v(t) dt\). 2. Add the constant of integration: \(x(t) = \int v(t) dt + C\). 3. Use the initial position to find the constant 'C': \(C = x_0 - \int v(0) dt\). 4. Write the position function: \(x(t) = \int v(t) dt + x_0 - \int v(0) dt\).

Step-by-step solution

Integrate the velocity function

To find the position function, integrate the given velocity function with respect to time: $$ x(t) = \int v(t) dt. $$

Add the constant of integration

After integrating the velocity function, we need to add a constant of integration 'C': $$ x(t) = \int v(t) dt + C. $$

Use the initial position to find the constant 'C'

Since we are given the initial position of the object (\(x(t) = x_0\) when \(t = 0\)), we can use these values to solve for 'C'. Plug in the values of \(t = 0\) and \(x(0) = x_0\) into the position function: $$ x_0 = \int v(0) dt + C. $$ Now, solve for 'C': $$ C = x_0 - \int v(0) dt. $$

Write the position function

Finally, substitute the value of 'C' back into the position function to obtain the complete position function for the object: $$ x(t) = \int v(t) dt + x_0 - \int v(0) dt. $$ This is the position function of the object for \(t \geq 0\), given its velocity function and initial position.

Key concepts

Initial Position

The initial position of an object is the starting point from where we begin tracing the motion of the object. It acts as a baseline to determine the future positions when the velocity is known. Knowing the initial position helps in solving for unknown constants when integrating functions like velocity over time.

• Notation: The initial position is often denoted as \( x_0 \), representing the position of the object at time \( t = 0 \).
• Role in Position Function: This value is pivotal because it allows us to calculate the constant of integration, ensuring that the derived position function accurately describes the object's motion from the correct starting point.

Velocity Function

The velocity function \( v(t) \) represents how an object's velocity changes over time. It is critical in understanding and predicting an object's future positions.

• Definition: Velocity is the rate of change of position with time. It can be constant or vary, depending on the forces acting on the object.
• Relation to Position: By integrating the velocity function, we can find the position function, which shows how the position changes with time.
• Application: Knowing the velocity at different times helps us calculate the total distance traveled, providing insights into the object's dynamics.

Integration

Integration is a mathematical process that helps us find the position function from the velocity function. Specifically, it sums up infinite tiny changes in velocity over time to give us the resultant change in position.

• Process of Integration: When we integrate the velocity function \( \int v(t) dt \), we're essentially accumulating the velocity over a time interval to determine the change in position.
• Physical Interpretation: Just like adding small pieces to get a whole, integration combines all small positional changes dictated by the velocity to provide an overall picture.
• Calculating Position: Integrating the velocity function gives us \( x(t) \), the position function, up to a constant of integration that still needs determining.

Constant of Integration

The constant of integration \( C \) emerges when we integrate the velocity function to derive the position function. It accounts for initial conditions and ensures an accurate representation of the object’s initial state.

• Need for the Constant: After integration, the velocity function turns into a position function that lacks specific starting point information, which the constant provides.
• Finding the Constant: We use the initial position \( x_0 \) to determine \( C \) by setting \( x(0) = x_0 \) and solving for \( C \).
• Substituting Back: Once \( C \) is found, it is added to the derived integral of velocity, \( \int v(t) dt \), resulting in a complete equation that describes the full motion \( x(t) = \int v(t) dt + C \).