---
title: "A drone is initially at rest at the origin of an \\(xy\\)-coordinate system. At time \\(t=0\\), the drone’s engines are engaged, giving it an acceleration vector \\(\\vec{a}(t)\\) as a function of time \\(t\\) given by the equation: \\[ \\vec{a}(t) = (\\alpha t) \\hat{i} + (\\beta – \\gamma t) \\hat{j} \\] where \\(\\alpha = 1.2 \\text{ m/s}^3\\), \\(\\beta = 4.0 \\text{ m/s}^2\\), and \\(\\gamma = 0.6 \\text{ m/s}^3\\)."
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url: "https://nerd-notes.com/ubq/117769/"
date_modified: "2026-08-04T07:56:17+00:00"
---

# A drone is initially at rest at the origin of an \(xy\)-coordinate system. At time \(t=0\), the drone’s engines are engaged, giving it an acceleration vector \(\vec{a}(t)\) as a function of time \(t\) given by the equation:
\[ \vec{a}(t) = (\alpha t) \hat{i} + (\beta – \gamma t) \hat{j} \]
where \(\alpha = 1.2 \text{ m/s}^3\), \(\beta = 4.0 \text{ m/s}^2\), and \(\gamma = 0.6 \text{ m/s}^3\).

A drone is initially at rest at the origin of an \(xy\)-coordinate system. At time \(t=0\), the drone's engines are engaged, giving it an acceleration vector \(\vec{a}(t)\) as a function of time \(t\) given by the equation:
\[ \vec{a}(t) = (\alpha t) \hat{i} + (\beta - \gamma t) \hat{j} \]
where \(\alpha = 1.2 \text{ m/s}^3\), \(\beta = 4.0 \text{ m/s}^2\), and \(\gamma = 0.6 \text{ m/s}^3\).

**Part a)** **Derive** an expression for the velocity vector of the drone, \(\vec{v}(t)\), as a function of time \(t\). Express your answer algebraically in terms of \(\alpha\), \(\beta\), \(\gamma\), \(t\), and fundamental constants, as appropriate. *(2 points)*

**Part b)** **Derive** an expression for the position vector of the drone, \(\vec{r}(t)\), as a function of time \(t\). Express your answer algebraically in terms of \(\alpha\), \(\beta\), \(\gamma\), \(t\), and fundamental constants, as appropriate. *(2 points)*

**Part c)** A student observing the drone claims that at \(t = 2.0 \text{ s}\), the drone is traveling at an angle greater than \(45^\circ\) above the \(+x\)-axis. *(3 points)*

**Part d)** **Calculate** the magnitude of the displacement of the drone from \(t = 0\) to \(t = 5.0 \text{ s}\). *(2 points)*

**Part e)** **Calculate** the average acceleration vector of the drone from \(t = 0\) to \(t = 5.0 \text{ s}\). Express your answer in unit vector notation. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117769/*