---
title: "A block of mass \\(M\\) moves to the right along a horizontal, frictionless surface with an initial speed \\(v_0\\) at time \\(t = 0\\). Starting at \\(t = 0\\), a time-dependent net horizontal force \\(F(t) = C t^2\\) is applied to the block in the direction of motion, where \\(C\\) is a positive constant. Which of the following integral equations correctly relates the speed \\(v(t)\\) of the block at a later time \\(t\\) to the force acting on it?"
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url: "https://nerd-notes.com/ubq/120727/"
date_modified: "2026-08-23T04:42:52+00:00"
---

# A block of mass \(M\) moves to the right along a horizontal, frictionless surface with an initial speed \(v_0\) at time \(t = 0\). Starting at \(t = 0\), a time-dependent net horizontal force \(F(t) = C t^2\) is applied to the block in the direction of motion, where \(C\) is a positive constant. Which of the following integral equations correctly relates the speed \(v(t)\) of the block at a later time \(t\) to the force acting on it?

A block of mass \(M\) moves to the right along a horizontal, frictionless surface with an initial speed \(v_0\) at time \(t = 0\). Starting at \(t = 0\), a time-dependent net horizontal force \(F(t) = C t^2\) is applied to the block in the direction of motion, where \(C\) is a positive constant. Which of the following integral equations correctly relates the speed \(v(t)\) of the block at a later time \(t\) to the force acting on it?

![A rectangular block labeled M rests on a horizontal solid line representing a frictionless surface. A horizontal arrow pointing to the right labeled v_0 is positioned directly above the block. A horizontal arrow labeled F(t) = C t^2 originates at the left vertical face of the block and points to the right. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460172-N1KW1X.jpg)

- **A.** \(\displaystyle \int_{v_0}^{v(t)} dv' = \dfrac{C}{M} \int_{0}^{t} t'^2 \, dt'\)
- **B.** \(\displaystyle \int_{0}^{v(t)} dv' = \dfrac{C}{M} \int_{0}^{t} t'^2 \, dt'\)
- **C.** \(\displaystyle \int_{v_0}^{v(t)} v' \, dv' = \dfrac{C}{M} \int_{0}^{t} t'^2 \, dt'\)
- **D.** \(\displaystyle \int_{v_0}^{v(t)} dv' = C M \int_{0}^{t} t'^2 \, dt'\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120727/*