9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

d. A piecewise function for s(t)

Verified step by step guidance

1. Identify the intervals on the time axis where the velocity function v(t) changes its behavior. From the graph, these intervals are: 0 ≤ t ≤ 2, 2 ≤ t ≤ 4, and 4 ≤ t ≤ 6.
2. For each interval, determine the equation of the velocity function v(t). For example, from t=0 to t=2, the velocity decreases linearly from 2 to 0, so find the slope and write the linear equation for v(t) in that interval. Repeat this for the other intervals.
3. Recall that the position function s(t) is the integral of the velocity function v(t) with respect to time t, plus the initial position s(0). Since s(0) = 0, the position function on each interval is the integral of the corresponding velocity function starting from the beginning of that interval, plus the position at the start of the interval.
4. Calculate the position function s(t) piecewise by integrating each velocity function over its interval. For each interval, express s(t) as an integral of v(t) plus the position value at the start of that interval (which comes from the previous interval's calculation).
5. Combine these results to write the full piecewise function for s(t), ensuring continuity at the interval boundaries by using the position values found at the end of each previous interval as the starting values for the next.

Key Concepts

Relationship Between Velocity and Position

Velocity is the derivative of position with respect to time, meaning that the position function s(t) can be found by integrating the velocity function v(t). Given an initial position s(0), integrating v(t) over time gives the displacement, which when added to s(0) yields the position at any time t.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. Since the velocity graph changes slope and sign at specific times, the position function s(t) must be expressed as a piecewise integral of v(t) over those intervals, ensuring continuity and correct initial conditions.

Definite Integration and Area Under the Curve

The position change over an interval is the definite integral of velocity over that interval, which corresponds to the net area under the velocity curve. Positive areas increase position, while negative areas decrease it, so calculating these areas accurately is essential for constructing s(t).