---
title: "A small cart of mass \\(m\\) is constrained to move along a straight, horizontal, frictionless track that aligns with the positive x-axis. The cart is initially at rest at the origin \\(x = 0\\). A stationary magnetic launcher located at the origin exerts a repulsive force on the cart in the positive x-direction. The magnitude of this force as a function of position \\(x\\) is given by \\(F(x) = F_0 e^{-\\alpha x}\\), where \\(F_0\\) and \\(\\alpha\\) are positive constants."
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url: "https://nerd-notes.com/ubq/117768/"
date_modified: "2026-08-04T07:56:11+00:00"
---

# A small cart of mass \(m\) is constrained to move along a straight, horizontal, frictionless track that aligns with the positive x-axis. The cart is initially at rest at the origin \(x = 0\). A stationary magnetic launcher located at the origin exerts a repulsive force on the cart in the positive x-direction. The magnitude of this force as a function of position \(x\) is given by \(F(x) = F_0 e^{-\alpha x}\), where \(F_0\) and \(\alpha\) are positive constants.

A small cart of mass \(m\) is constrained to move along a straight, horizontal, frictionless track that aligns with the positive x-axis. The cart is initially at rest at the origin \(x = 0\). A stationary magnetic launcher located at the origin exerts a repulsive force on the cart in the positive x-direction. The magnitude of this force as a function of position \(x\) is given by \(F(x) = F_0 e^{-\alpha x}\), where \(F_0\) and \(\alpha\) are positive constants.

![A horizontal straight line representing a track. On the left end of the track sits a stationary vertical rectangular block labeled 'Launcher'. Immediately to the right of the launcher, a rectangular cart with a label '\(m\)' sits on the track. A horizontal dashed line is positioned below the track to represent an axis. On this dashed line, a vertical tick mark aligned with the left side of the cart is labeled '\(x = 0\)'. Further to the right on the dashed line is another vertical tick mark labeled '\(x = d\)'. Exactly one arrow appears: it points horizontally to the right, starting at the center of the cart and ending slightly to the right of the cart. The text '\(F(x)\)' is placed just above this arrow. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830171-LXR9Sp.jpg)

**Part a)** **Derive** an expression for the work done by the magnetic launcher on the cart as it moves from \(x = 0\) to a position \(x = d\). Express your answer in terms of \(F_0\), \(\alpha\), \(d\), and fundamental constants, as appropriate. *(2 points)*

**Part b)** **Derive** an expression for the speed of the cart when it reaches the position \(x = d\). Express your answer in terms of \(m\), \(F_0\), \(\alpha\), \(d\), and fundamental constants, as appropriate. *(2 points)*

**Part c)** The instantaneous power delivered to the cart by the launcher changes as the cart moves along the track. *(3 points)*

**Part d)** Assuming the force exerted by the launcher is conservative and the potential energy of the cart-launcher system is zero at \(x = 0\), **derive** an expression for the potential energy \(U(x)\) of the system as a function of position \(x\). Express your answer in terms of \(F_0\), \(\alpha\), \(x\), and fundamental constants, as appropriate. *(3 points)*

**Part e)** **Determine** the maximum speed the cart achieves as it travels a very large distance away from the launcher (as \(x \to \infty\)). Express your answer in terms of \(m\), \(F_0\), \(\alpha\), and fundamental constants, as appropriate. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117768/*