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    <header><h1 class="page-title">Gravity Through Contradiction</h1></header>
    <address><time datetime="2022-03-23">March 23, 2022</time></address>

    <section class="abstract">
        <p class="eyebrow">A thought experiment about falling bodies</p>
        <p class="lead">Tie a light body to a heavy one, and the rule “heavier falls faster” predicts that the pair must fall both slower and faster than the heavy body alone.</p>
        <p>That contradiction breaks the rule before any object leaves the bench. Two shoes and a piece of string reveal the structure of gravity more clearly than a race from a tower.</p>
    </section>

    <main>
        <section>
            <h2>Join The Two Falling Bodies</h2>
            <p>Place a heavy red shoe and a light white shoe at the same height. Assume the red shoe falls faster.</p>
            <p>Now join them with a loose string. Under the assumed rule, the slower white shoe must hold the red shoe back, so the pair should fall more slowly than the red shoe alone.</p>
            <p>But the pair now weighs more than the red shoe. The same rule therefore says that the pair should fall faster.</p>
            <p>One connection forces the rule to assign opposite outcomes to the same body under the same conditions. The string does not measure gravity; it exposes a contradiction inside the proposed law.</p>
        </section>

        <section>
            <h2>The Rule Defeats Itself</h2>
            <p>Put the reasoning in order:</p>
            <ol>
                <li>Assume a heavier body falls faster than a lighter body.</li>
                <li>Join a light body to a heavy body.</li>
                <li>Because the light body is slower, it should retard the heavy body. The pair should fall more slowly than the heavy body alone.</li>
                <li>Because the pair is heavier than the heavy body alone, the original rule also says the pair should fall faster.</li>
                <li>The same pair cannot be both slower and faster than the same comparison body, in the same conditions and at the same time.</li>
                <li>Therefore the starting rule is inconsistent.</li>
            </ol>
            <p>The decisive move follows the assumed rule until it produces two incompatible answers. No timing measurement can repair a rule that contradicts itself.</p>
            <p>This is a Galileo-style <em>reductio ad absurdum</em>: assume the claim, derive its contradiction, and reject the starting assumption. The argument uses the principle of noncontradiction, distinct from the law of excluded middle.</p>
        </section>

        <section>
            <h2>Air Explains The Leaf And The Apple</h2>
            <p>A broad leaf often reaches the ground after a compact apple because every falling object pushes air aside and the air pushes back. Shape, area, orientation, speed, and density determine the drag force. A leaf flutters, rotates, and repeatedly presents new area; an apple cuts a steadier path.</p>
            <p>Shape alone can transform the result. A flat sheet of paper descends slowly. Crumple that same sheet into a ball and it falls much faster through the room even though its mass barely changes.</p>
            <p>Remove the air and those aerodynamic differences disappear. At the same location in an ideal vacuum, bodies with the same initial motion share the same gravitational acceleration regardless of mass. Near Earth, that acceleration stays approximately constant across modest height differences.</p>
            <p>The joined-body argument isolates gravitational acceleration; ordinary falling motion combines gravity with drag, buoyancy, wind, shape, and rotation.</p>
        </section>

        <section>
            <h2>Mass Cancels in the Acceleration</h2>
            <p>The same result appears in the familiar equations.</p>
            <p>Near Earth’s surface, the gravitational force on an object with gravitational mass <var>m<sub>g</sub></var> is approximately:</p>
            <pre><code>F = m_g g</code></pre>
            <p>Newton’s second law relates net force to inertial mass <var>m<sub>i</sub></var> and acceleration:</p>
            <pre><code>F = m_i a</code></pre>
            <p>If gravity is the only significant force, combine them:</p>
            <pre><code>m_i a = m_g g
      a = (m_g / m_i) g</code></pre>
            <p>Measurements show gravitational and inertial mass proportional with extraordinary precision. In the usual units their ratio equals one, so <code>a = g</code>. A larger mass feels a larger gravitational force and requires proportionally more force for the same acceleration; the two mass factors cancel.</p>
            <p>Air resistance scales differently, which lets leaves, feathers, raindrops, shoes, and apples follow different paths while sharing the same mass-independent gravitational acceleration.</p>
        </section>

        <section>
            <h2>A Real Observation Can Support The Wrong Rule</h2>
            <p>Stones often beat leaves to the ground. The observation holds; the quick inference fails because several causes share the visible motion.</p>
            <p>Gravity, drag, buoyancy, wind, shape, and rotation all contribute. Naming the most conspicuous property as the sole cause can produce a rule that works in familiar cases and collapses when those cases combine.</p>
            <p>The joined-body argument goes beyond a surprising counterexample. It shows that the proposed rule cannot assign a coherent outcome to a system made from the very bodies it claims to describe.</p>
            <p>The method travels beyond mechanics: compose two cases, check the whole made from the parts, and follow every consequence. A sound rule must still produce one coherent answer after composition.</p>
        </section>

        <section>
            <h2>The String Makes Gravity Legible</h2>
            <p>An apple and a leaf still reach the ground at different times in the park because air acts on their shapes differently. In a vacuum, they keep pace.</p>
            <p>The string changes neither object’s gravitational nature. It reveals that mass-dependent gravitational acceleration cannot survive the act of joining the objects.</p>
            <p>A simple composition has turned a familiar observation into a deeper law: gravity gives every freely falling body the same local acceleration.</p>
        </section>
    </main>

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        <p>Next:</p>
        <a href="/linkedin-archive/2020-05-18-do-not-skip-variational-calculus.html">Do Not Skip Variational Calculus</a>
        <p>Continue from one falling-body argument into the mathematical language for comparing possible motions and paths.</p>
        <p>More:</p>
        <a href="/from-the-ground-up/logic/">Core Logic</a>
        <p>Explore the Boolean operations and reasoning structures used to make contradictions and alternatives explicit.</p>
        <a href="/">Back to greenforest.io</a>
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