Gravitational Fall: Does a Dropped Probe Keep Speeding Up?
1 · Predict
A probe is released from rest and falls straight toward a central mass under inverse-square gravity. As it drops from radius 3.0 m to 2.0 m to 1.0 m, how does its speed change?
- It gains speed at a steady rate, just like a ball dropped near Earth's surface.
- It gains speed faster and faster — the pull grows stronger as it gets closer.
- It slows down as it approaches, because the central mass repels fast-movers.
2 · Set Up
- Open the grav-fall preset. A single probe sits at rest a distance r₀ = 3.5 m from the central mass at the center of the canvas.
- Confirm the r-probe (distance) and speed probe are both attached to the probe body before you run anything.
- Run the simulation and pause it as the probe passes r = 3.0 m, then 2.0 m, then 1.0 m, reading the speed probe at each radius.
3 · Collect Data
| Release Radius r₀ (m) | Radius r (m) | Escape Speed at r (m/s) | Fall Speed (m/s) |
|---|---|---|---|
| 3.5 | 3 | ||
| 3.5 | 2 | ||
| 3.5 | 1 |
Plot fall speed (y-axis) against radius r (x-axis). Note that r decreases as the probe falls, so read the curve right-to-left.
4 · Analyze
- The probe falls the same 0.5 m from 3.5→3.0 m and again from 1.5→1.0 m, yet it gains far more speed over the inner interval. Use your data to explain why equal distances do not give equal speed gains here (unlike free fall at constant g).
- For each row, compare your measured fall speed to the escape speed at that same radius. The fall speed is always smaller — connect this to the fact that the probe started from rest (its total energy is negative, so it stays bound).
5 · Extend
- Near Earth's surface we use v = √(2g·Δh) with a constant g. Explain why that formula fails here, and what quantity replaces the constant g as the probe moves inward.
- The idealized formula v = √(2GM(1/r − 1/r₀)) blows up to infinity as r → 0. The real simulation instead pins the probe just outside the center. What physical reason (and what modeling choice) keeps the speed finite?
The Physics Behind This Experiment
Gravitational Potential Energy
As the probe falls inward, r shrinks and this (negative) potential energy becomes more negative, releasing energy that reappears as kinetic energy — which is why the probe keeps speeding up rather than falling at a steady rate.
Escape Speed
The speed a body would need at radius r to just barely escape the central mass. A probe dropped from rest starts with negative total energy, so its fall speed stays below this value at every radius — it is bound, not escaping.