---
title: "A horizontal conveyor belt moves at a constant speed \\(v_0\\). Sand is dropped vertically from rest onto the belt at a time-dependent rate given by \\(\\dfrac{dm}{dt} = kt\\), where \\(k\\) is a positive constant and \\(t\\) is time. Friction between the sand and the belt quickly accelerates the sand horizontally to match the speed of the belt. Which of the following expressions represents the total work done by the motor driving the belt during the time interval from \\(t = 0\\) to \\(t = T\\)?"
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url: "https://nerd-notes.com/ubq/117671/"
date_modified: "2026-08-04T07:49:46+00:00"
---

# A horizontal conveyor belt moves at a constant speed \(v_0\). Sand is dropped vertically from rest onto the belt at a time-dependent rate given by \(\dfrac{dm}{dt} = kt\), where \(k\) is a positive constant and \(t\) is time. Friction between the sand and the belt quickly accelerates the sand horizontally to match the speed of the belt. Which of the following expressions represents the total work done by the motor driving the belt during the time interval from \(t = 0\) to \(t = T\)?

A horizontal conveyor belt moves at a constant speed \(v_0\). Sand is dropped vertically from rest onto the belt at a time-dependent rate given by \(\dfrac{dm}{dt} = kt\), where \(k\) is a positive constant and \(t\) is time. Friction between the sand and the belt quickly accelerates the sand horizontally to match the speed of the belt. Which of the following expressions represents the total work done by the motor driving the belt during the time interval from \(t = 0\) to \(t = T\)?

![A horizontal conveyor belt supported by two circular rollers moves to the right at constant velocity vector v_0. Above the left portion of the belt, a funnel-shaped hopper releases a stream of sand falling vertically downward onto the top surface of the belt. A small layer of sand rests on the belt, moving to the right at velocity vector v_0. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-diagram-1-1785829786-tvza90.jpg)

- **A.** \(\dfrac{1}{4} k v_0^2 T^2\)
- **B.** \(\dfrac{1}{3} k v_0^2 T^2\)
- **C.** \(\dfrac{1}{2} k v_0^2 T^2\)
- **D.** \(k v_0^2 T^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117671/*