HKDSE • Compulsory Part: Force and Motion • Circular motion & gravitation

Newton's Cannon & Satellite Orbits

Drag the launch-speed slider to fire a cannonball from a high mountain. Watch it fall to Earth, skim into an elliptical orbit, settle into a circular orbit, or escape — the same gravity that makes apples fall also keeps satellites aloft.

Observation focus · As you raise the launch speed, watch the path change from hitting the ground, to an ellipse, to a circle, then to escape.

Parameters

Exaggerated mountain for Newton's thought experiment; GEO ≈ 35 786 km.

Simulation runs faster than real time so orbits fit on screen.

Presets

Challenge

Tune $v$ (near circular) until the cannonball completes one full revolution without hitting Earth.

Waiting for a closed orbit…

Live readings

Speed $v$ 7.00 km/s
Radial distance $r$ 7171 km
Gravity $F=GMm/r^2$ 7.6 kN
Accel. $a=GM/r^2$ 7.6 m/s²
Orbital period $T$ —
Eccentricity $e$ 0.00
Orbit type Falling
Circular $v_c=\sqrt{GM/r}$ 7.45 km/s
Escape $v_{esc}=\sqrt{2GM/r}$ 10.54 km/s

HKDSE formulae

$$ F = \dfrac{GMm}{r^2} $$
$$ v_c = \sqrt{\dfrac{GM}{r}} $$

Why? Gravity always pulls toward Earth's centre. If the sideways speed is just right, the curved path of the fall matches Earth's curvature — so the projectile keeps missing the ground and becomes a satellite.

Time is sped up on screen — see the time-scale control.

Newton's mountain
×400
Drag to rotate • pinch / scroll to zoom
Gravity $F$ Velocity $v$ Acceleration $a$ Orbit trail

Physics note

Newton imagined a cannon on a tall mountain. At low $v$ the ball hits the ground nearby; raise $v$ and the landing point moves farther around the globe. When $v = \sqrt{GM/r}$, the path is a closed circle — a satellite. Between circular and escape speeds the orbit is an ellipse; above $v_{esc}=\sqrt{2GM/r}$ the ball never returns. Geostationary satellites sit much higher ($h\approx 36\,000\,\mathrm{km}$) so their period matches one sidereal day.