---
title: "A block of mass \\(m\\) rests on a horizontal, frictionless surface. At time \\(t = 0\\), a time-dependent horizontal force given by \\(F(t) = ct^2\\), where \\(c\\) is a positive constant, is applied to the block in the positive \\(x\\)-direction."
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url: "https://nerd-notes.com/ubq/117759/"
date_modified: "2026-08-04T07:56:04+00:00"
---

# A block of mass \(m\) rests on a horizontal, frictionless surface. At time \(t = 0\), a time-dependent horizontal force given by \(F(t) = ct^2\), where \(c\) is a positive constant, is applied to the block in the positive \(x\)-direction.

A block of mass \(m\) rests on a horizontal, frictionless surface. At time \(t = 0\), a time-dependent horizontal force given by \(F(t) = ct^2\), where \(c\) is a positive constant, is applied to the block in the positive \(x\)-direction.

![A rectangular block labeled 'm' sitting on a horizontal flat line representing a frictionless surface. A horizontal arrow labeled 'F(t)' points to the right, originating from the center of the right side of the block. A small standard xy-coordinate axis is lightly shown to the side, with +x pointing to the right. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830164-7UpB9B.jpg)

**Part a)** On the dot below, which represents the block, **draw** and label the forces (not components) that act on the block at a time \(t > 0\). Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. *(2 points)*

**Part b)** **Derive** an expression for the acceleration \(a(t)\) of the block as a function of time \(t\). Express your answer in terms of \(m\), \(c\), \(t\), and fundamental constants. *(2 points)*

**Part c)** Using calculus, **derive** an expression for the velocity \(v(t)\) of the block as a function of time \(t\). Express your answer in terms of \(m\), \(c\), \(t\), and fundamental constants. *(2 points)*

**Part d)** A student performs an experiment with the block using the values \(m = 2.0 \text{ kg}\) and \(c = 1.5 \text{ N/s}^2\). **Calculate** the time required for the block to reach a velocity of \(16 \text{ m/s}\). *(2 points)*

**Part e)** The experiment is repeated using a heavier block of mass \(M = 16 \text{ kg}\) while keeping the same force constant \(c = 1.5 \text{ N/s}^2\). **Calculate** the new time required for this block to reach a velocity of \(16 \text{ m/s}\). *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117759/*