---
title: "A motorized cart is constrained to move along a straight horizontal track that aligns with the \\(x\\)-axis. The position of the cart as a function of time \\(t\\) for \\(t \\ge 0\\) is given by the equation \\(x(t) = b t^3 – c t^2\\), where \\(b\\) and \\(c\\) are positive constants."
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url: "https://nerd-notes.com/ubq/117756/"
date_modified: "2026-08-04T07:55:59+00:00"
---

# A motorized cart is constrained to move along a straight horizontal track that aligns with the \(x\)-axis. The position of the cart as a function of time \(t\) for \(t \ge 0\) is given by the equation \(x(t) = b t^3 – c t^2\), where \(b\) and \(c\) are positive constants.

A motorized cart is constrained to move along a straight horizontal track that aligns with the \(x\)-axis. The position of the cart as a function of time \(t\) for \(t \ge 0\) is given by the equation \(x(t) = b t^3 - c t^2\), where \(b\) and \(c\) are positive constants.

**Part a)** Using calculus, **derive** expressions for the velocity \(v(t)\) and acceleration \(a(t)\) of the cart as functions of time \(t\). Express your answers in terms of \(b\), \(c\), and \(t\). *(2 points)*

**Part b)** **Calculate** the time \(t_1\) at which the cart momentarily comes to rest (other than at \(t=0\)). Express your answer in terms of \(b\) and \(c\). *(2 points)*

**Part c)** **Sketch** the graphs of velocity \(v\) versus time \(t\) and acceleration \(a\) versus time \(t\) for the cart from \(t = 0\) to a time \(t > t_1\). Explicitly **label** the time \(t_1\) on the horizontal axis of each graph. *(4 points)*

**Part d)** **Justify** the shape of your velocity graph from \(t=0\) to \(t=t_1\) by relating it to the acceleration of the cart during this time interval. *(2 points)*

**Part e)** **Indicate** whether the speed of the cart is increasing, decreasing, or not changing at the exact instant \(t = \dfrac{c}{3b}\). - [ ] Increasing - [ ] Decreasing - [ ] Not changing **Justify** your answer using the expressions you derived in part (a). *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117756/*