17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.


a. Determine the position function, for t≥0, using the antiderivative method


v(t) = 6−2t on [0, 5]; s(0)=0

Verified step by step guidance

1. Identify the given velocity function: \(v(t) = 6 - 2t\) and the initial position \(s(0) = 0\).
2. Recall that the position function \(s(t)\) is the antiderivative (indefinite integral) of the velocity function \(v(t)\), so write \(s(t) = \int v(t) \, dt + C\).
3. Integrate the velocity function: compute \(\int (6 - 2t) \, dt\) by integrating each term separately.
4. After integration, express \(s(t)\) as \(s(t) = 6t - t^2 + C\), where \(C\) is the constant of integration.
5. Use the initial condition \(s(0) = 0\) to solve for \(C\) by substituting \(t=0\) into \(s(t)\) and setting it equal to zero.

Key Concepts

Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. The position function can be found by integrating the velocity function over time, which accumulates the changes in position from the initial point.

Antiderivative (Indefinite Integral)

The antiderivative of a function is another function whose derivative is the original function. Finding the antiderivative of velocity gives the position function up to a constant, which can be determined using initial conditions.

Initial Conditions and Constants of Integration

When integrating to find position from velocity, an unknown constant appears. The initial position value (e.g., s(0) = 0) is used to solve for this constant, ensuring the position function accurately reflects the object's starting point.