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title: "A particle is launched horizontally with initial speed \\(v_0\\) from the top of a cliff of height \\(H\\) at time \\(t = 0\\). The particle moves under the influence of constant downward gravitational acceleration of magnitude \\(g\\). In addition, a horizontal wind exerts a force producing a time-dependent horizontal acceleration \\(a_x(t) = -b t\\), where \\(b\\) is a positive constant and the positive \\(x\\)-direction is aligned with the initial velocity. Which of the following expressions represents the horizontal position \\(x\\) of the particle when it lands at the base of the cliff (\\(y = 0\\))?"
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url: "https://nerd-notes.com/ubq/117588/"
date_modified: "2026-08-04T07:49:13+00:00"
---

# A particle is launched horizontally with initial speed \(v_0\) from the top of a cliff of height \(H\) at time \(t = 0\). The particle moves under the influence of constant downward gravitational acceleration of magnitude \(g\). In addition, a horizontal wind exerts a force producing a time-dependent horizontal acceleration \(a_x(t) = -b t\), where \(b\) is a positive constant and the positive \(x\)-direction is aligned with the initial velocity. Which of the following expressions represents the horizontal position \(x\) of the particle when it lands at the base of the cliff (\(y = 0\))?

A particle is launched horizontally with initial speed \(v_0\) from the top of a cliff of height \(H\) at time \(t = 0\). The particle moves under the influence of constant downward gravitational acceleration of magnitude \(g\). In addition, a horizontal wind exerts a force producing a time-dependent horizontal acceleration \(a_x(t) = -b t\), where \(b\) is a positive constant and the positive \(x\)-direction is aligned with the initial velocity. Which of the following expressions represents the horizontal position \(x\) of the particle when it lands at the base of the cliff (\(y = 0\))?

![A vertical two-dimensional coordinate system showing a vertical cliff face on the left at x = 0 extending from y = 0 up to height H. A particle is located at (0, H) with a horizontal arrow labeled v_0 pointing to the right, a vertical arrow labeled g pointing downward, and a horizontal arrow labeled a_x = -bt pointing to the left. A dashed parabolic arc curve extends from (0, H) downward and to the right, landing on the horizontal ground at coordinate (x, 0). The cliff height H is indicated with a vertical dimension line. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829753-x8K9IU.jpg)

- **A.** \(\sqrt{\dfrac{2H}{g}} \left( v_0 - \dfrac{b H}{g} \right)\)
- **B.** \(\sqrt{\dfrac{2H}{g}} \left( v_0 - \dfrac{2b H}{3g} \right)\)
- **C.** \(\sqrt{\dfrac{2H}{g}} \left( v_0 - \dfrac{b H}{3g} \right)\)
- **D.** \(\sqrt{\dfrac{2H}{g}} \left( v_0 + \dfrac{b H}{3g} \right)\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117588/*