Choosing the Appropriate Gravitational Potential Formula & Handling Significant Figures

Problem Context and Setup

• Coordinate choice: Speaker labels the horizontal axis as the x-axis.
  • Gravitational force vector is explicitly said to be "pointing this way," i.e.
    downward or toward the Earth’s center, perpendicular to the defined x-axis.
• Goal of the problem: determine which energy terms belong in an energy-conservation calculation for a projectile (or object) that moves only a small distance above Earth’s surface.

Selecting Relevant Energy Terms

• Explicit checklist offered by the speaker:
  • Spring (Elastic) Potential Energy — not used because there is no spring in the scenario (“we can just eliminate these, those springs”).
  • Rotational / Spinning Kinetic Energy — not used (“Are we gonna be using spinning energy in this problem? Are we spinning? … No.”).
• Remaining candidate: Gravitational Potential Energy.
  • Decision hinges on how far the object travels vertically.

Two Forms of Gravitational Potential Energy

1. Near-surface (linear) approximation U_{\text{near}} = mgh
   • $m$ = mass (kg)
   • $g \approx 9.80\,\text{m/s}^2$
   • $h$ = height above reference level (m)
   • Valid when $h \ll R_E$ (Earth’s radius).
2. Universal (exact) form U{\text{exact}} = -\frac{G ME m}{r}
   • $G = 6.67\times10^{-11}\,\text{N·m}^2!/\text{kg}^2$
   • $M_E = 5.97\times10^{24}\,\text{kg}$
   • $r$ = distance from Earth’s center (m).
   • Must be used when $h$ is comparable to $R_E$ or for orbital motion.

Why the Near-Surface Approximation Is Acceptable Here

• Numerical clues: the object’s furthest radial coordinate becomes r \approx 6.3\,\text{Mm} to 6.37\,\text{Mm} ("6.3" vs "6.37").
  • Earth’s mean radius R_E \approx 6.38\,\text{Mm} (more precisely 6.378\,\text{Mm}).
• The change in $r$ is only about \Delta r \approx 0.07\,\text{Mm} = 70\,\text{km}, which is roughly 1\% of $R_E$.
  • Because \frac{\Delta r}{R_E} \approx 10^{-2}\, (\text{small}), the linear mgh model produces negligible error for typical exam/ homework accuracy requirements.

Precision & Significant Figures (Sig Figs)

• To bolster accuracy, keep extra digits on both the initial and final radii when using the exact form:
  • Suggestion: “take three or four sig figs here … four or five sig figs here.”
• Example workflow for the exact formula:
  1. Record initial radius r_i = 6.378\,\text{Mm} to 5 sig figs.
  2. Record final radius r_f = 6.307\,\text{Mm} (hypothetical) to 5 sig figs.
  3. Compute \Delta U = -\frac{GME m}{rf} + \frac{GME m}{ri} using 5–6 sig figs internally.
  4. Round the final answer only to the course-required sig figs (usually 2–3).
• If instead you use mgh with h = rf - ri, any rounding error in $h$ propagates less severely, but the same rule—retain guard digits until the end—applies.

Practical Advice for Problem Solving

• Build an energy inventory before writing the equation. Ask:
  1. Does the object stretch/compress a spring?
     → If not, remove $\frac{1}{2}kx^2$.
  2. Is there rotation or "spinning" of the body?
     → If not, remove $\frac{1}{2}I\omega^2$.
  3. Is there translation?
     → Keep $\frac{1}{2}mv^2$ if speed is non-zero.
  4. Is gravitational potential relevant?
     → Yes, choose mgh or -GMm/r based on height scale.
• Always label axes so force directions and sign conventions stay clear.

Common Pitfalls & How to Avoid Them

• Over-precision vs. under-precision: rounding too early can disguise conceptual errors; keeping a few guard digits prevents loss of accuracy.
• Inconsistent potentials: don’t mix mgh for one height and -GMm/r for another within the same equation; choose one form consistently.
• Forgetting reference level: in mgh, $h$ is measured from a chosen zero-potential surface; in the universal form, $U=0$ is at r \to \infty by definition.

Ethical / Philosophical Connection

• The instructor emphasizes judgment: physics isn’t just plugging numbers; it’s choosing the correct model. This highlights a scientist’s ethical duty to recognize approximations and their limitations.

Real-World Relevance

• Satellite launches, ballistic missiles, and space tourism all require choosing between mgh and the exact gravitational potential; using the wrong model can cost millions.
• Engineering tolerances depend on proper sig-fig handling; over-rounding can compromise safety factors.