Motion of a ball rolled over a shallow step

Keith Zengel; Laruen Boehnert

arxiv: 2605.20481 · v1 · pith:3Y63YPA6new · submitted 2026-05-19 · ⚛️ physics.ed-ph

Motion of a ball rolled over a shallow step

Keith Zengel , Laruen Boehnert This is my paper

Pith reviewed 2026-05-21 06:06 UTC · model grok-4.3

classification ⚛️ physics.ed-ph

keywords ball rollingshallow steptrajectory deflectionno-slip conditionvelocity increasephysics demonstrationclassical mechanics

0 comments

The pith

Rolling a ball over a shallow step increases its velocity perpendicular to the edge and deflects its path.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a ball rolled over a shallow step gains speed in the direction across the step when it maintains no-slip contact with the edge. This added perpendicular velocity component changes the ball's direction once it reaches the lower surface. The authors derive the equations governing this motion and confirm the deflection through a simple experiment that uses only a ball and a stack of papers. The shallow step height relative to the ball radius keeps the motion visible and prevents bouncing, making the effect easy to observe in a classroom setting.

Core claim

Under the assumption that the ball rolls without slipping throughout its contact with the step edge, the motion produces an increase in the velocity component perpendicular to the step. This increase deflects the ball's trajectory after it contacts the lower platform. The paper presents the derived equations for the ball's motion in this no-slip case and reports experimental results that match the predicted deflection.

What carries the argument

The no-slip rolling condition while the ball is in contact with the step edge, which transfers rotational effects into a net gain in perpendicular translational velocity.

If this is right

The ball's path bends away from its initial straight line after dropping to the lower level.
The magnitude of the perpendicular velocity increase depends on the height of the step relative to the ball radius.
The effect remains observable without bouncing when the step is shallow compared to the ball size.
The same setup can be used to demonstrate cases where speed is maintained or reduced if slipping occurs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar velocity changes might appear in rolling objects crossing small obstacles on surfaces used in sports or robotics.
Extending the model to include partial slipping could predict a wider range of possible deflections or speed changes.
The classroom demonstration suggests a low-cost way to explore conservation principles during brief edge contact.

Load-bearing premise

The ball rolls without slipping the entire time it is in contact with the step edge.

What would settle it

Measure the outgoing direction or perpendicular velocity component of the ball after it crosses the step and check whether the observed deflection matches the value predicted by the derived equations for the given step height and ball radius.

read the original abstract

A ball rolled over a shallow step will experience an increase in velocity along the direction perpendicular to the step. This causes a deflection in the ball's trajectory. In this paper we derive the equations that describe the motion of a ball rolled over a shallow step and present the results of our experimental test. This simple demonstration can be used in any classroom where the physics teacher has access to a ball and a stack of papers. Prior work has shown that a ball rolled over an edge can maintain its speed, as is commonly assumed, but it can also experience an increase or even decrease in speed. The ball can either roll without slipping while it is in contact with the edge, or else begin to slip before it leaves the edge. In this paper we will consider the case where the ball rolls without slipping the entire time it is in contact with the step edge, then contacts a lower platform. We work with shallow step heights relative to the radius of the ball so that the motion of the ball is easy to observe at all times, and so that the ball does not bounce when it encounters the lower platform. These shallow step heights mean that we can assume the ball does not slip as it moves over the edge.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a perpendicular velocity gain and trajectory deflection for no-slip rolling over a shallow step, with a simple classroom test, but the result hinges on an assumption that may not hold under realistic friction.

read the letter

The main takeaway is that a ball rolled without slipping over a shallow step picks up a sideways velocity component that deflects its path, and the authors work out the equations for that case while running a basic experiment with a ball and stacked papers. They build directly on earlier studies of balls at edges that can gain or lose speed depending on slip, but isolate the no-slip scenario for the shallow geometry. The shallow-height limit keeps the motion visible and prevents bounces, which makes the setup practical for a classroom demo. The derivation rests on standard rigid-body constraints at the contact point, and the experiment aims to confirm the predicted deflection. That is the useful part: a concrete, low-cost illustration of how an impulsive contact under no-slip changes the velocity vector. The work is honest about extending published results rather than claiming an entirely new phenomenon. The central soft spot is the no-slip assumption itself. The paper justifies it by the shallow step making contact gentle, yet the edge interaction delivers an impulsive normal force. Maintaining zero velocity at the contact point requires friction to deliver a precise impulse; if the available friction falls short, slipping begins and the kinematic relations used for the perpendicular velocity no longer apply. Earlier work already shows that slipping can produce either speed gain or loss, so the no-slip prediction is not automatic. The abstract gives no friction estimates, high-speed checks, or error analysis, which leaves the experimental support hard to judge. If the full paper includes those details and shows the ball really stayed in no-slip throughout contact, the claim strengthens. This is aimed at physics teachers or lab instructors who need a hands-on rolling-mechanics activity. A reader hunting for major new mechanics results will not find them, but someone assembling classroom examples of constraint forces will see direct value. It is solid enough on its own terms to merit peer review rather than a desk reject, mainly so referees can check the assumption validation and the quantitative match between derivation and data.

Referee Report

2 major / 2 minor

Summary. The paper claims that a ball rolled over a shallow step without slipping gains a velocity component perpendicular to the step edge, producing a measurable deflection in its subsequent trajectory. The authors derive the relevant kinematic and dynamic equations under the assumptions of no slip, shallow step height relative to ball radius, and no bounce upon landing, then report experimental tests of the predicted deflection using a simple classroom setup with a ball and stacked papers. Prior literature on slipping cases is noted, with the present work restricted to the sustained no-slip regime.

Significance. If the no-slip condition is validated and the deflection is quantitatively confirmed, the result supplies a low-cost, visually accessible demonstration of how impulsive contact forces at an edge can redistribute velocity components in rolling without slipping. This could strengthen introductory treatments of rigid-body mechanics and friction by showing a case where speed is not conserved along the original direction.

major comments (2)

[Abstract, final paragraph] Abstract, final paragraph: the central claim of a perpendicular velocity increase rests on the assumption that the ball rolls without slipping for the entire duration of edge contact. The shallow-height argument is offered as justification, yet the contact involves an impulsive normal force whose friction requirement is not shown to remain below the static limit for the materials used; without this check or direct experimental confirmation (e.g., high-speed imaging or post-contact surface marks), the kinematic relations used to obtain the perpendicular component may not apply.
[Experimental results section] Experimental results section: the abstract states that experimental tests are presented, but the manuscript must supply quantitative comparison (measured deflection angle versus predicted value, with uncertainties) rather than qualitative agreement; otherwise the support for the derived velocity change remains inconclusive.

minor comments (2)

[Introduction] Define the shallow-step regime quantitatively (e.g., explicit bound on h/R) so that readers can judge the range of validity without ambiguity.
[Experimental methods] Add a brief statement of the coefficient of friction or surface preparation used in the experiments to allow reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address the major comments point by point below and have made revisions to improve the manuscript's clarity and rigor.

read point-by-point responses

Referee: [Abstract, final paragraph] Abstract, final paragraph: the central claim of a perpendicular velocity increase rests on the assumption that the ball rolls without slipping for the entire duration of edge contact. The shallow-height argument is offered as justification, yet the contact involves an impulsive normal force whose friction requirement is not shown to remain below the static limit for the materials used; without this check or direct experimental confirmation (e.g., high-speed imaging or post-contact surface marks), the kinematic relations used to obtain the perpendicular component may not apply.

Authors: We thank the referee for highlighting this important point regarding the no-slip assumption. To strengthen the manuscript, we will add a section analyzing the friction force required during the impulsive contact. Using the derived equations for the velocity change and the geometry of the shallow step, we calculate the minimum coefficient of friction needed to prevent slipping and compare it to literature values for the ball-paper interface. For the shallow heights considered (much less than the ball radius), the required friction is modest and within typical static friction limits. We will also note that the absence of observed slipping or bouncing in experiments supports this. While we did not perform high-speed imaging, the consistency between theory and observed deflections provides indirect validation. This addition will be included in the revised theoretical and experimental sections. revision: yes
Referee: [Experimental results section] Experimental results section: the abstract states that experimental tests are presented, but the manuscript must supply quantitative comparison (measured deflection angle versus predicted value, with uncertainties) rather than qualitative agreement; otherwise the support for the derived velocity change remains inconclusive.

Authors: We agree that a quantitative comparison is essential for conclusive support. In the revised manuscript, we will present the experimental data quantitatively, including measured deflection angles from multiple trials, the predicted values from the model, and associated uncertainties derived from measurement errors in step height, ball radius, and angle determination. A table or graph will show the agreement within uncertainties, thereby providing stronger evidence for the velocity change. We have performed this analysis on our existing data and will incorporate it into the experimental results section. revision: yes

Circularity Check

0 steps flagged

Derivation uses standard rigid-body mechanics under an explicit no-slip input assumption

full rationale

The paper states it derives the perpendicular velocity change and trajectory deflection from rigid-body kinematics under the maintained no-slip condition during edge contact. This no-slip condition is introduced as an assumption justified by the shallow step height relative to ball radius, not extracted from or defined by the final velocity or deflection results. No equations are presented that equate a fitted parameter to a predicted ratio, no self-citation supplies a uniqueness theorem, and no ansatz is smuggled in. The chain therefore remains self-contained against external Newtonian mechanics benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation rests on standard Newtonian mechanics plus two domain assumptions stated in the abstract; no free parameters or new entities are introduced in the provided text.

axioms (2)

domain assumption The ball rolls without slipping the entire time it is in contact with the step edge
Explicitly stated in the final paragraph of the abstract as the regime considered.
domain assumption Shallow step height relative to ball radius prevents bouncing and keeps motion observable
Stated in the abstract to justify the modeling choices.

pith-pipeline@v0.9.0 · 5738 in / 1212 out tokens · 45602 ms · 2026-05-21T06:06:38.496490+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

v + ωR = 0 (rolling without slipping)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

+12𝑚𝑣!"+12𝑚𝑣#

Wentworth Institute of Technology, Boston, MA. email: zengelk@wit.edu 2022-08-03 Introduction A ball rolled over a shallow step will experience an increase in velocity along the direction perpendicular to the step. This causes a deflection in the ball’s trajectory. In this paper we derive the equations that describe the motion of a ball rolled over a shal...

work page 2022
[2]

(18) and 𝐵p=C1−ℎ(1+𝛽)𝑅D

FIG. 2: A sketch of the experimental setup shown from the top-down view (a) and the side view (b). Finally, we transferred the video files to Tracker5 for analysis. A screenshot from Tracker5 in Fig. 3 shows the path of the ball as it moves from the upper platform to the lower platform. FIG. 3: A screenshot from Tracker showing the tracked path of the bal...

work page 2004

[1] [1]

+12𝑚𝑣!"+12𝑚𝑣#

Wentworth Institute of Technology, Boston, MA. email: zengelk@wit.edu 2022-08-03 Introduction A ball rolled over a shallow step will experience an increase in velocity along the direction perpendicular to the step. This causes a deflection in the ball’s trajectory. In this paper we derive the equations that describe the motion of a ball rolled over a shal...

work page 2022

[2] [2]

(18) and 𝐵p=C1−ℎ(1+𝛽)𝑅D

FIG. 2: A sketch of the experimental setup shown from the top-down view (a) and the side view (b). Finally, we transferred the video files to Tracker5 for analysis. A screenshot from Tracker5 in Fig. 3 shows the path of the ball as it moves from the upper platform to the lower platform. FIG. 3: A screenshot from Tracker showing the tracked path of the bal...

work page 2004