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.

c. The position at t=5

Verified step by step guidance

1. Understand that the position function \(s(t)\) is the integral of the velocity function \(v(t)\) over time, starting from the initial position \(s(0) = 0\).
2. Identify the intervals on the graph where the velocity function changes. From the graph, note the velocity values and the time intervals: from \(t=0\) to \(t=2\), velocity decreases linearly from 2 to -2; from \(t=2\) to \(t=3\), velocity increases linearly from -2 to 3; from \(t=3\) to \(t=5\), velocity decreases linearly from 3 to 0.
3. Calculate the area under the velocity curve for each interval, as the integral of velocity over time gives the change in position. Since the graph consists of straight lines, these areas can be found by calculating the areas of geometric shapes (triangles and trapezoids) formed between the velocity curve and the time axis.
4. Sum the areas from each interval, taking care to consider the sign of the velocity (areas below the time axis represent negative displacement). This sum will give the net change in position from \(t=0\) to \(t=5\).
5. Add the net change in position to the initial position \(s(0) = 0\) to find the position at \(t=5\), \(s(5)\).

Key Concepts

Velocity and Position Relationship

Velocity is the rate of change of position with respect to time. To find the position at a given time, you integrate the velocity function over the time interval, starting from the initial position. This means the position is the accumulated area under the velocity-time graph.

Definite Integral as Area Under the Curve

The definite integral of a velocity function from time a to b represents the net displacement during that interval. Graphically, this corresponds to the area between the velocity curve and the time axis, where areas above the axis are positive and below are negative.

Piecewise Linear Functions and Area Calculation

When the velocity graph is piecewise linear, the area under the curve can be found by calculating areas of simple geometric shapes like triangles and rectangles. Summing these areas with appropriate signs gives the net displacement, which is used to find the position.