How can a satellite keep falling without reaching the ground?
Without gravity, would it move radially outward or along the tangent?
- Start a circular orbit and show vectors.
- Pause and advance one step at a time.
- Compare how velocity and acceleration turn.
What to measure
Compare x, y, v_x, v_y and |a| at two positions.
The idea underneath
Velocity carries the body along a tangent while gravity bends its path inward. Apparent weightlessness is shared free fall, not the absence of gravity. In circular motion acceleration changes direction without changing speed.
a = −μr/|r|³; a_c = v²/r; v·a = 0 (circular)r is position vector in m; a in m/s²; v in m/s.
Build the argument
- Start from F = GMm/r² and divide by satellite mass: a = μ/r².
- For circular motion set v²/r = μ/r². Then v = √(μ/r) and T = 2π√(r³/μ).
- Combine kinetic and potential energy per mass: ε = v²/2 − μ/r. Its sign separates bound and unbound trajectories.
Worked example 1
At r=7×10⁶ m, acceleration is 8.13 m/s², far from zero.
Worked example 2
In 1 s the small inward fall is approximately aΔt²/2=4.07 m while tangential travel is about 7.55 km.
Applications
Orbital mechanics connects satellite navigation, communications and interplanetary transfers. A launch is a choice of position and velocity, not a command to follow a circle.
Common misconception
Zero weight is not zero gravity. Orbital radius is measured from the centre. Numerical energy drift must not be mistaken for real energy production.
A further investigation
Use a dot product to check perpendicular velocity and acceleration.
Check your understanding
Why do orbiting occupants feel weightless?
- There is no gravity
- Spacecraft and contents fall together
Explanation
Shared free fall removes the usual support force. In a circular orbit v·a≈0.
Model limits
Newtonian two-body model with fixed central mass. No drag, third bodies, oblateness or relativity. Visual sizes are illustrative; initial conditions are fictional, not live ephemerides.
University Physics Volume 1, 13.2 Gravitation Near Earth’s Surface
University Physics Volume 1, 13.4 Satellite Orbits and Energy