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title: "A cargo spacecraft performs an in-line docking maneuver with a space station along an axis where the station’s docking port is fixed at \\(x = 0\\). The spacecraft approaches from the positive \\(x\\)-direction, and its velocity \\(v(t) = \\dfrac{dx}{dt}\\) relative to the station is shown in the graph: a constant velocity of \\(-4\\text{ m/s}\\) from \\(t = 0\\text{ s}\\) to \\(t = 4\\text{ s}\\), a linear change to \\(+2\\text{ m/s}\\) at \\(t = 10\\text{ s}\\), a linear change to \\(-2\\text{ m/s}\\) at \\(t = 14\\text{ s}\\), and a linear return to \\(0\\text{ m/s}\\) at \\(t = 18\\text{ s}\\). The spacecraft successfully latches to the docking port at \\(x = 0\\) at time \\(t = 18\\text{ s}\\). What was the initial separation distance between the spacecraft and the docking port at \\(t = 0\\text{ s}\\)?"
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url: "https://nerd-notes.com/ubq/124079/"
date_modified: "2026-09-28T13:30:21+00:00"
---

# A cargo spacecraft performs an in-line docking maneuver with a space station along an axis where the station’s docking port is fixed at \(x = 0\). The spacecraft approaches from the positive \(x\)-direction, and its velocity \(v(t) = \dfrac{dx}{dt}\) relative to the station is shown in the graph: a constant velocity of \(-4\text{ m/s}\) from \(t = 0\text{ s}\) to \(t = 4\text{ s}\), a linear change to \(+2\text{ m/s}\) at \(t = 10\text{ s}\), a linear change to \(-2\text{ m/s}\) at \(t = 14\text{ s}\), and a linear return to \(0\text{ m/s}\) at \(t = 18\text{ s}\). The spacecraft successfully latches to the docking port at \(x = 0\) at time \(t = 18\text{ s}\). What was the initial separation distance between the spacecraft and the docking port at \(t = 0\text{ s}\)?

A cargo spacecraft performs an in-line docking maneuver with a space station along an axis where the station's docking port is fixed at \(x = 0\). The spacecraft approaches from the positive \(x\)-direction, and its velocity \(v(t) = \dfrac{dx}{dt}\) relative to the station is shown in the graph: a constant velocity of \(-4\text{ m/s}\) from \(t = 0\text{ s}\) to \(t = 4\text{ s}\), a linear change to \(+2\text{ m/s}\) at \(t = 10\text{ s}\), a linear change to \(-2\text{ m/s}\) at \(t = 14\text{ s}\), and a linear return to \(0\text{ m/s}\) at \(t = 18\text{ s}\). The spacecraft successfully latches to the docking port at \(x = 0\) at time \(t = 18\text{ s}\). What was the initial separation distance between the spacecraft and the docking port at \(t = 0\text{ s}\)?

![A quantitative Cartesian coordinate graph with a horizontal axis labeled t (s) and a vertical axis labeled v (m/s). The horizontal axis has major gridlines and numeric tick labels at intervals of 2 from 0 to 20. The vertical axis has major gridlines and numeric tick labels at intervals of 2 from -6 to 4, with a prominent solid horizontal axis line at v = 0. A single continuous thick solid black line depicts the piecewise function: starting at (0, -4), running horizontally to (4, -4); then connecting with a straight line segment to (10, 2), crossing the horizontal axis at (8, 0); then connecting with a straight line segment to (14, -2), crossing the horizontal axis at (12, 0); and finally connecting with a straight line segment to (18, 0) where it terminates. Solid circular dots mark the vertices at (0, -4), (4, -4), (10, 2), (14, -2), and (18, 0). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602221-xrFU3J.jpg)

- **A.** \(18\text{ m}\)
- **B.** \(26\text{ m}\)
- **C.** \(30\text{ m}\)
- **D.** \(34\text{ m}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124079/*