---
title: "A crate of mass \\(m\\) rests on a rough horizontal floor. The coefficient of kinetic friction between the crate and the floor is \\(\\mu_k\\). A student applies a horizontal force to the right to push the crate. The graph below shows the velocity \\(v\\) of the crate as a function of time \\(t\\). The motion is divided into three distinct intervals. In the interval from \\(t_2\\) to \\(t_3\\), the student stops pushing the crate entirely, and it coasts to a stop."
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url: "https://nerd-notes.com/ubq/114495/"
date_modified: "2026-07-03T04:38:48+00:00"
---

# A crate of mass \(m\) rests on a rough horizontal floor. The coefficient of kinetic friction between the crate and the floor is \(\mu_k\). A student applies a horizontal force to the right to push the crate. The graph below shows the velocity \(v\) of the crate as a function of time \(t\). The motion is divided into three distinct intervals. In the interval from \(t_2\) to \(t_3\), the student stops pushing the crate entirely, and it coasts to a stop.

A crate of mass \(m\) rests on a rough horizontal floor. The coefficient of kinetic friction between the crate and the floor is \(\mu_k\). A student applies a horizontal force to the right to push the crate. The graph below shows the velocity \(v\) of the crate as a function of time \(t\). The motion is divided into three distinct intervals. In the interval from \(t_2\) to \(t_3\), the student stops pushing the crate entirely, and it coasts to a stop.

![A line graph with velocity \(v\) on the vertical axis and time \(t\) on the horizontal axis. The vertical axis has a positive tick mark labeled \(v_0\). The horizontal axis has three positive tick marks labeled \(t_1\), \(t_2\), and \(t_3\). The graph starts at the origin (0,0), goes up as a straight diagonal line with a positive slope to the point \((t_1, v_0)\). From \(t_1\) to \(t_2\), the graph is a horizontal line at \(v = v_0\). From \(t_2\) to \(t_3\), the graph goes down as a straight diagonal line with a negative slope, ending at the point \((t_3, 0)\).](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1783053528-Lu6Z8g.jpg)

**Part a)** The dots below represent the crate during the three time intervals. On each dot, **draw and label** the forces (not components) that are exerted on the crate during that interval. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot. The relative lengths of all arrows should indicate the relative magnitudes of the forces. *(4 points)*

**Part b)** **Explain** how the features of the velocity-time graph provide evidence for the relative magnitudes of the horizontal forces you drew for each of the three intervals.

**Part c)** On the axes below, **sketch** a graph of the acceleration \(a\) of the crate as a function of time \(t\) between \(t=0\) and \(t=t_3\).

**Part d)** Suppose instead that during Interval 2 (between \(t_1\) and \(t_2\)), a block of mass \(m\) is gently dropped onto the crate. After the block lands, the student continues to push the crate with the exact same constant horizontal force as they did before the block was dropped. **Indicate** the direction of the acceleration of the crate-block system immediately after the block lands. - [ ] In the direction of motion - [ ] Opposite to the direction of motion - [ ] The acceleration is zero **Justify** your answer using physical principles.


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/114495/*