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title: "A spacecraft of total mass \\(M\\) is at rest in an inertial reference frame in deep space. The spacecraft consists of a main capsule and an attached fuel canister connected by an internal spring mechanism that stores elastic potential energy \\(U_0\\). When the mechanism is released, the spring expands and pushes the canister away along a straight line, converting all stored energy \\(U_0\\) into the kinetic energy of the separating capsule and canister.  Three different canister designs are evaluated, each using an identical spring mechanism with the same initial stored energy \\(U_0\\):  Design I: The canister has mass \\(m_1 = \\dfrac{1}{4}M\\), leaving a capsule of mass \\(\\dfrac{3}{4}M\\).  Design II: The canister has mass \\(m_2 = \\dfrac{1}{2}M\\), leaving a capsule of mass \\(\\dfrac{1}{2}M\\).  Design III: The canister has mass \\(m_3 = \\dfrac{3}{4}M\\), leaving a capsule of mass \\(\\dfrac{1}{4}M\\).  Which of the following correctly ranks the final kinetic energy of the main capsule, \\(K_1\\), \\(K_2\\), and \\(K_3\\), in the original reference frame after separation for the three designs?"
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url: "https://nerd-notes.com/ubq/124380/"
date_modified: "2026-09-28T14:05:04+00:00"
---

# A spacecraft of total mass \(M\) is at rest in an inertial reference frame in deep space. The spacecraft consists of a main capsule and an attached fuel canister connected by an internal spring mechanism that stores elastic potential energy \(U_0\). When the mechanism is released, the spring expands and pushes the canister away along a straight line, converting all stored energy \(U_0\) into the kinetic energy of the separating capsule and canister.

Three different canister designs are evaluated, each using an identical spring mechanism with the same initial stored energy \(U_0\):

Design I: The canister has mass \(m_1 = \dfrac{1}{4}M\), leaving a capsule of mass \(\dfrac{3}{4}M\).

Design II: The canister has mass \(m_2 = \dfrac{1}{2}M\), leaving a capsule of mass \(\dfrac{1}{2}M\).

Design III: The canister has mass \(m_3 = \dfrac{3}{4}M\), leaving a capsule of mass \(\dfrac{1}{4}M\).

Which of the following correctly ranks the final kinetic energy of the main capsule, \(K_1\), \(K_2\), and \(K_3\), in the original reference frame after separation for the three designs?

A spacecraft of total mass \(M\) is at rest in an inertial reference frame in deep space. The spacecraft consists of a main capsule and an attached fuel canister connected by an internal spring mechanism that stores elastic potential energy \(U_0\). When the mechanism is released, the spring expands and pushes the canister away along a straight line, converting all stored energy \(U_0\) into the kinetic energy of the separating capsule and canister.

Three different canister designs are evaluated, each using an identical spring mechanism with the same initial stored energy \(U_0\):

Design I: The canister has mass \(m_1 = \dfrac{1}{4}M\), leaving a capsule of mass \(\dfrac{3}{4}M\).

Design II: The canister has mass \(m_2 = \dfrac{1}{2}M\), leaving a capsule of mass \(\dfrac{1}{2}M\).

Design III: The canister has mass \(m_3 = \dfrac{3}{4}M\), leaving a capsule of mass \(\dfrac{1}{4}M\).

Which of the following correctly ranks the final kinetic energy of the main capsule, \(K_1\), \(K_2\), and \(K_3\), in the original reference frame after separation for the three designs?

![A horizontal schematic of the spacecraft system before separation. On the left is a larger rectangular block labeled Main Capsule. On the right is a smaller rectangular block labeled Canister. Between the adjacent vertical faces of the two blocks is a horizontal coiled spring shown in compression. A horizontal dashed centerline passes through the centers of both blocks. Above the spring, a centered label indicates U_0. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604303-A3v4ho.jpg)

- **A.** \(K_3 > K_2 > K_1\)
- **B.** \(K_1 > K_2 > K_3\)
- **C.** \(K_2 > K_1 = K_3\)
- **D.** \(K_1 = K_2 = K_3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124380/*