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title: "A train travels along a straight, horizontal track aligned with the \\(+x\\)-axis with a time-dependent acceleration given by \\(a_x(t) = bt\\), where \\(b\\) is a positive constant. The train is at rest at time \\(t = 0\\). Due to horizontal wind shear, the velocity of falling raindrops relative to the ground is \\(\\vec{v}_{r/g}(t) = (u_0 + ct)\\hat{i} – v_0\\hat{j}\\), where \\(u_0\\), \\(c\\), and \\(v_0\\) are positive constants, and \\(\\hat{j}\\) points vertically upward. In terms of the given quantities, at what time \\(t > 0\\) do the raindrops appear to fall purely vertically to an observer on the train?"
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url: "https://nerd-notes.com/ubq/120511/"
date_modified: "2026-08-23T04:39:19+00:00"
---

# A train travels along a straight, horizontal track aligned with the \(+x\)-axis with a time-dependent acceleration given by \(a_x(t) = bt\), where \(b\) is a positive constant. The train is at rest at time \(t = 0\). Due to horizontal wind shear, the velocity of falling raindrops relative to the ground is \(\vec{v}_{r/g}(t) = (u_0 + ct)\hat{i} – v_0\hat{j}\), where \(u_0\), \(c\), and \(v_0\) are positive constants, and \(\hat{j}\) points vertically upward. In terms of the given quantities, at what time \(t > 0\) do the raindrops appear to fall purely vertically to an observer on the train?

A train travels along a straight, horizontal track aligned with the \(+x\)-axis with a time-dependent acceleration given by \(a_x(t) = bt\), where \(b\) is a positive constant. The train is at rest at time \(t = 0\). Due to horizontal wind shear, the velocity of falling raindrops relative to the ground is \(\vec{v}_{r/g}(t) = (u_0 + ct)\hat{i} - v_0\hat{j}\), where \(u_0\), \(c\), and \(v_0\) are positive constants, and \(\hat{j}\) points vertically upward. In terms of the given quantities, at what time \(t > 0\) do the raindrops appear to fall purely vertically to an observer on the train?

![A horizontal track line spans the bottom of the diagram. At the lower-left, a pair of perpendicular coordinate axes has a horizontal arrow labeled \(\hat{i}\) pointing right and a vertical arrow labeled \(\hat{j}\) pointing upward. A single rectangular cart representing a train with two circular wheels rests on the track. A horizontal arrow pointing to the right above the cart is labeled \(a_x(t) = bt\). In the upper-right area, three identical small solid dots representing raindrops are aligned diagonally, each having an attached arrow pointing downward and rightward at an angle labeled \(\vec{v}_{r/g}\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787459958-L1XwL1.jpg)

- **A.** \(t = \dfrac{c + \sqrt{c^2 + 4bu_0}}{2b}\)
- **B.** \(t = \dfrac{c + \sqrt{c^2 + 2bu_0}}{b}\)
- **C.** \(t = \dfrac{-c + \sqrt{c^2 + 2bu_0}}{b}\)
- **D.** \(t = \dfrac{c + \sqrt{c^2 + bu_0}}{b}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120511/*