---
title: "A block of mass \\(m\\) is initially at rest on a horizontal surface where the coefficient of kinetic friction between the block and the surface is \\(\\mu\\). Starting at time \\(t = 0\\), a constant horizontal pushing force of magnitude \\(F_P\\) is applied to the block, causing it to accelerate to the right. At time \\(t = t_1\\), the pushing force \\(F_P\\) is abruptly removed. The block continues to slide to the right until it comes to rest at time \\(t = t_2\\)."
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url: "https://nerd-notes.com/ubq/109199/"
date_modified: "2026-04-02T01:44:47+00:00"
---

# A block of mass \(m\) is initially at rest on a horizontal surface where the coefficient of kinetic friction between the block and the surface is \(\mu\). Starting at time \(t = 0\), a constant horizontal pushing force of magnitude \(F_P\) is applied to the block, causing it to accelerate to the right. At time \(t = t_1\), the pushing force \(F_P\) is abruptly removed. The block continues to slide to the right until it comes to rest at time \(t = t_2\).

A block of mass \(m\) is initially at rest on a horizontal surface where the coefficient of kinetic friction between the block and the surface is \(\mu\). Starting at time \(t = 0\), a constant horizontal pushing force of magnitude \(F_P\) is applied to the block, causing it to accelerate to the right. At time \(t = t_1\), the pushing force \(F_P\) is abruptly removed. The block continues to slide to the right until it comes to rest at time \(t = t_2\).

![A square block resting on a horizontal surface represented by a horizontal line. An arrow pointing to the right starts at the left edge of the block, labeled 'F_P'. The mass of the block is labeled 'm' inside the square. A surface texture or hash marks below the horizontal line indicate a rough surface.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1774089788-UrQr8v.jpg)

**Part a)** **Draw** and **label** the forces (not components) that act on the block during the time interval \(0 < t < t_1\). 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 speed of the block \(v_1\) at time \(t = t_1\). Express your answer in terms of \(m\), \(\mu\), \(F_P\), \(t_1\), and fundamental constants as appropriate. *(3 points)*

**Part c)** **Derive** an expression for the total time \(t_2\) that the block is in motion (from the moment it starts moving until it comes to rest). Express your answer in terms of \(m\), \(\mu\), \(F_P\), \(t_1\), and fundamental constants as appropriate. *(3 points)*

**Part d)** A second, identical experiment is performed, but the original block is replaced with a block of mass \(2m\). The same force \(F_P\) is applied for the same time interval \(t_1\). Assume the new block also successfully moves to the right during this interval. **Predict** whether the new total time in motion, \(t_{new}\), is greater than, less than, or equal to the original total time \(t_2\). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your prediction using the functional dependence found in your derivation from part (c). *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/109199/*