---
title: "A model rocket cart of initial mass \\( M_0 \\) rests on a straight, horizontal, frictionless track. At time \\( t = 0 \\), the rocket engine ignites and ejects gas backwards at a constant rate \\( R \\) (where \\( R = \\dfrac{dm_{gas}}{dt} \\)). The gas is ejected with a constant speed \\( v_e \\) relative to the rocket cart. The engine operates until a total mass \\( M_g \\) of gas has been ejected."
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url: "https://nerd-notes.com/ubq/117775/"
date_modified: "2026-08-04T07:56:37+00:00"
---

# A model rocket cart of initial mass \( M_0 \) rests on a straight, horizontal, frictionless track. At time \( t = 0 \), the rocket engine ignites and ejects gas backwards at a constant rate \( R \) (where \( R = \dfrac{dm_{gas}}{dt} \)). The gas is ejected with a constant speed \( v_e \) relative to the rocket cart. The engine operates until a total mass \( M_g \) of gas has been ejected.

A model rocket cart of initial mass \( M_0 \) rests on a straight, horizontal, frictionless track. At time \( t = 0 \), the rocket engine ignites and ejects gas backwards at a constant rate \( R \) (where \( R = \dfrac{dm_{gas}}{dt} \)). The gas is ejected with a constant speed \( v_e \) relative to the rocket cart. The engine operates until a total mass \( M_g \) of gas has been ejected.

![A rectangular cart labeled \( M_0 \) rests on a horizontal line representing a track. On the left side of the cart, a nozzle is drawn with horizontal dashed lines emerging to the left, representing ejected gas. An arrow labeled \( v \) points to the right from the cart, indicating its direction of motion. An arrow labeled \( v_e \) points to the left from the ejected gas, indicating the relative velocity of the exhaust. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830197-XWoelJ.jpg)

**Part a)** **Determine** an expression for the mass \( m(t) \) of the rocket cart as a function of time \( t \) while the engine is operating. Express your answer in terms of \( M_0 \), \( R \), \( t \), and fundamental constants as appropriate. *(1 points)*

**Part b)** Consider a small time interval \( dt \) during which the rocket cart, moving at speed \( v \), ejects a small mass \( dm \). *(3 points)*

**Part c)** Using the differential equation from part (b)(ii), **derive** an expression for the speed \( v(t) \) of the rocket cart as a function of time \( t \). Express your answer in terms of \( M_0 \), \( R \), \( v_e \), \( t \), and fundamental constants as appropriate. *(3 points)*

**Part d)** **Determine** an expression for the acceleration of the rocket cart at the instant the engine shuts off. Express your answer in terms of \( M_0 \), \( M_g \), \( R \), \( v_e \), and fundamental constants as appropriate. *(2 points)*

**Part e)** A second, identical rocket cart is tested. The second rocket's engine ejects gas at a constant rate \( 2R \), but with a constant exhaust speed of \( \dfrac{v_e}{2} \). The second engine also operates until a total mass \( M_g \) of gas has been ejected. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117775/*