---
title: "A passenger inside a train car accelerating horizontally to the right with constant acceleration \\(a_0\\) releases a small sphere from rest relative to the ceiling. A stationary observer on the ground outside the train watches the sphere fall through a transparent window. The observer inside the train records the trajectory of the sphere as a straight line tilted backward, whereas the observer on the ground records the trajectory as a parabola. Which of the following correctly explains why the two observers record different trajectory shapes for the falling sphere?"
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url: "https://nerd-notes.com/ubq/120505/"
date_modified: "2026-08-23T04:39:15+00:00"
---

# A passenger inside a train car accelerating horizontally to the right with constant acceleration \(a_0\) releases a small sphere from rest relative to the ceiling. A stationary observer on the ground outside the train watches the sphere fall through a transparent window. The observer inside the train records the trajectory of the sphere as a straight line tilted backward, whereas the observer on the ground records the trajectory as a parabola. Which of the following correctly explains why the two observers record different trajectory shapes for the falling sphere?

A passenger inside a train car accelerating horizontally to the right with constant acceleration \(a_0\) releases a small sphere from rest relative to the ceiling. A stationary observer on the ground outside the train watches the sphere fall through a transparent window. The observer inside the train records the trajectory of the sphere as a straight line tilted backward, whereas the observer on the ground records the trajectory as a parabola. Which of the following correctly explains why the two observers record different trajectory shapes for the falling sphere?

![A rectangular train car sits on two circular wheels on a flat horizontal track line. A single horizontal arrow above the train roof points to the right and is labeled \(a_0\). Inside the train car, a small solid black circle representing a sphere is positioned near the ceiling. A stick figure representing the passenger stands inside the train car to the right of the sphere. Outside the train car on the left, a second stick figure representing the ground observer stands on the ground. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787459955-illfdW.jpg)

- **A.** The interior observer records a straight line because the sphere maintains a constant horizontal velocity relative to the car while accelerating downward, whereas the ground observer records a parabola because the ground frame accelerates relative to the sphere.
- **B.** The ground observer records a parabola because the sphere experiences both non-zero horizontal and vertical accelerations relative to the ground, whereas the interior observer records a straight line because the train acceleration cancels the sphere horizontal displacement.
- **C.** The interior observer records a straight line because the constant ratio of relative acceleration components ensures equal horizontal and vertical displacements at all times, whereas the ground observer records a parabola because the horizontal velocity of the sphere continuously changes after release.
- **D.** In the ground frame, the sphere has a non-zero initial horizontal velocity and zero horizontal acceleration, making horizontal displacement linear in time and vertical displacement quadratic in time; in the train frame, the sphere starts from rest and both horizontal and vertical displacements are quadratic in time under constant relative acceleration components.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120505/*