Orbits and Escape Velocities: Newton's Cannonball

Put a cannon on top of a very high mountain and fire horizontally, faster each time. A slow cannonball lands near the mountain, a faster one lands farther away. At some speed the planet's surface curves away beneath the ball exactly as fast as the ball falls — and it never comes down. That is the first cosmic velocity v₁, and the ball moves in a circle. Fire faster and the orbit stretches into an ellipse; at the escape velocity v₂ = √2·v₁ the body leaves the planet for ever. The dashed line predicts where the ball will go; press “Newton's fan” to see every case at once, and turn on “equal areas” to test Kepler's second law.

falls onto the planet circle (v₀ = v₁) ellipse parabola (v₀ = v₂) hyperbola (v₀ > v₂) predicted path →v— velocity →F— gravity (towards the planet’s centre)

The cannon fires horizontally. Gravity always points to the planet’s centre and turns the velocity; the F arrow grows as 1/r² when the ball comes closer. There is no air and the planet does not rotate — just as in Newton’s thought experiment.

Planet

h height of the mountain 400

v₀ launch speed 7.30

v₁v₂

Playback speed

Predicted path (dashed)

Vectors v and F

Equal areas (Kepler’s 2nd law)

The “= v₁” and “= v₂” buttons set the exact speed for the current height. Motion is computed from exact formulas (Kepler’s equation), so a circle never drifts and the playback speed does not change the numbers.

v₁ = √(GM/(R+h))

v₂ = √2·v₁

v₀ / v₁

e — eccentricity

periapsis altitude

apoapsis altitude

a — semi-major axis

T — period

E/m = v²/2 − GM/r

T²/a³

Solve the problem, enter the answer and press “Check” — the simulator will immediately show this case with a shot.

Flight graphs

Speed, altitude and energy of the ball versus time. The speed is greatest at periapsis and smallest at apoapsis. Kinetic and potential energy turn into each other while the total E stays constant: E < 0 — ellipse, E = 0 — parabola, E > 0 — hyperbola.

This thought experiment was invented by Isaac Newton himself. In “A Treatise of the System of the World” (published in 1728, after his death) he drew a mountain from which a stone is thrown harder and harder: it lands farther and farther away until it circles the Earth and comes back to the summit. This is how Newton explained why the Moon does not fall onto the Earth: it is falling all the time, but it moves sideways so fast that it keeps missing. Almost 230 years later Newton's drawing came true. On 4 October 1957 the first artificial satellite was launched from Baikonur in Kazakhstan, and on 12 April 1961 Yuri Gagarin flew from the same launch site: his Vostok circled the planet in 108 minutes. In January 1959 the Luna 1 probe became the first craft to reach escape velocity and left the Earth for ever, becoming an artificial planet of the Solar System.

KEY FORMULAS

First cosmic (orbital) velocity

$$\frac{GMm}{(R+h)^2} = \frac{mv_1^2}{R+h} \;\Rightarrow\; v_1 = \sqrt{\frac{GM}{R+h}}, \qquad v_1\big|_{h=0} = \sqrt{gR}$$

G = 6.67·10⁻¹¹ N·m²/kg² is the gravitational constant; M and R are the mass and radius of the planet; h is the height of the orbit above the surface; g = GM/R² is the free-fall acceleration at the surface; at the Earth's surface √(gR) ≈ 7.9 km/s

▶ Detailed Explanation

The first cosmic velocity is the speed at which a body launched horizontally moves round the planet in a circle. Gravity acts entirely as the centripetal force: it does not change the magnitude of the velocity, it only turns its direction. The result is uniform circular motion with radius r = R + h.

For the Earth (GM = 3.986·10¹⁴ m³/s², R = 6,371 km), at the surface v₁ = √(3.986·10¹⁴ / 6.371·10⁶) ≈ 7.91 km/s. At the ISS altitude of 400 km: v₁ = √(3.986·10¹⁴ / 6.771·10⁶) ≈ 7.67 km/s. The higher the orbit, the lower v₁: a satellite 36,000 km up (geostationary orbit) moves at only 3.07 km/s. The satellite's own mass does not appear in the formula — a light and a heavy satellite at the same height move identically.

Second cosmic (escape) velocity

$$\frac{mv_2^2}{2} - \frac{GMm}{R+h} = 0 \;\Rightarrow\; v_2 = \sqrt{\frac{2GM}{R+h}} = \sqrt{2}\,v_1$$

v₂ is the smallest speed at which a body leaves the planet for ever; at v = v₂ the path is a parabola, at v > v₂ it is a hyperbola. At the Earth's surface v₂ ≈ 11.2 km/s

▶ Detailed Explanation

The potential energy of a body in the planet's field is Ep = −GMm/r: it is negative and tends to zero far away. To escape, the total energy must be at least zero: mv²/2 − GMm/r ≥ 0. Hence v₂ = √(2GM/r).

The direction of the velocity does not matter here: whichever way you fire (as long as it is not into the planet), at v ≥ v₂ the body leaves. For the Earth at the surface v₂ ≈ 11.2 km/s, for the Moon 2.38 km/s, for Jupiter 60 km/s. If v > v₂, far from the planet the body keeps the speed v∞ = √(v² − v₂²) — this is how flights to other planets are planned.

Energy and the shape of the orbit

$$E = \frac{mv^2}{2} - \frac{GMm}{r} = -\frac{GMm}{2a}; \qquad E<0:\ e<1, \qquad E=0:\ e=1, \qquad E>0:\ e>1$$

a is the semi-major axis; e is the eccentricity of the orbit: e = 0 — circle, 0 < e < 1 — ellipse, e = 1 — parabola, e > 1 — hyperbola. The total energy is conserved

▶ Detailed Explanation

While the body flies, its kinetic and potential energies turn into each other, but their sum stays the same (a horizontal line on the energy graph). The sign of this sum decides the body's fate. At E < 0 it is bound to the planet and moves along an ellipse (a circle is a special case); at E = 0 along a parabola; at E > 0 along a hyperbola.

Conservation of energy gives the orbital speed formula (the vis-viva equation): v² = GM(2/r − 1/a). At periapsis r is small and the speed is greatest; at apoapsis it is smallest. For a circular orbit r = a, which gives v₁; as a → ∞ we get v₂.

Kepler's second law (law of equal areas)

$$\frac{\Delta S}{\Delta t} = \frac{1}{2}\,r\,v_\perp = \text{const}$$

ΔS is the area swept by the line from the planet's centre to the satellite in time Δt; v⊥ is the component of velocity perpendicular to that line

▶ Detailed Explanation

In a short time Δt the satellite moves by v·Δt, and the line from the planet's centre sweeps a thin triangle of area ½·r·v⊥·Δt. Gravity points to the centre and has no moment, so the angular momentum L = m·r·v⊥ is conserved — and therefore the rate of sweeping area is the same at every point of the orbit.

In the simulator the orbit is divided into 12 equal time intervals, with the sectors coloured alternately. At periapsis the satellite is fast and close — the sector is short and wide; at apoapsis it is slow and far — long and narrow. The areas measured from the drawn sectors equal ½·r₀·v₀·Δt.

Kepler's third law and the orbital period

$$\frac{T^2}{a^3} = \frac{4\pi^2}{GM}; \qquad e=0:\quad T = \frac{2\pi (R+h)}{v_1} = 2\pi\sqrt{\frac{(R+h)^3}{GM}}$$

T is the orbital period; a is the semi-major axis of the orbit (for a circular orbit, e = 0, a = R + h)

▶ Detailed Explanation

The squares of the orbital periods are proportional to the cubes of the semi-major axes. The ratio T²/a³ is the same for all satellites of one planet and depends only on its mass — so by measuring the Moon's period and orbital radius you can “weigh” the Earth.

Example: at 400 km (a = 6,771 km) T = 2π·√((6.771·10⁶)³ / 3.986·10¹⁴) ≈ 5,545 s ≈ 92 min. At 9 km/s from the same height a ≈ 10,850 km and T ≈ 3 h 07 min: the semi-major axis grew 1.6 times, the period 1.6^1.5 ≈ 2 times.

Why a satellite does not fall. Once the ball leaves the cannon, the only force on it is gravity, F = GMm/r², directed towards the centre of the planet (air resistance is ignored, as in Newton's thought experiment). So the ball is falling all the time. But the planet is round: while the ball moves sideways, the ground beneath it curves away. If the speed is high enough, the ball moves sideways so far during its fall that its height above the surface does not change. It is falling but can never land — that is orbital motion. Astronauts on the ISS are weightless precisely because they and the station are in continuous free fall.

First cosmic velocity. On a circular orbit of radius r = R + h gravity provides the centripetal acceleration: GMm/r² = mv²/r. Hence v₁ = √(GM/(R+h)). At the Earth's surface (h = 0) this is √(gR) ≈ 7.9 km/s; at the ISS altitude of 400 km it is 7.67 km/s, and one revolution takes about 92 minutes. The higher the orbit, the lower the speed needed and the longer the period. On the Moon v₁ is only 1.68 km/s, on Jupiter it is 42 km/s.

Speed below v₁. The ball follows part of an ellipse whose farthest point from the centre (apoapsis) is the launch point. If this ellipse crosses the surface, the ball lands: at v = 7.0 km/s from 400 km it lands after 12.6 minutes, having travelled 5,170 km; at 7.5 km/s it stays up for 28 minutes and covers 12,280 km. High above the planet, however, you can fly slower than v₁: from 2,000 km a ball at 6.8 km/s (v₁ there is 6.90 km/s) does not land — it dips to 1,530 km and comes back.

Between v₁ and v₂: an ellipse. Now the launch point is the closest point to the planet (periapsis), and the orbit stretches into an ellipse with the planet's centre at one focus (Kepler's first law). At 9 km/s from 400 km the satellite climbs to 8,560 km and comes back after 3 h 07 min. By conservation of energy, v²/2 − GM/r = const: moving away, the body slows down; coming closer, it speeds up.

Escape velocity. To leave for ever, the kinetic energy must be enough to overcome the attraction: mv²/2 ≥ GMm/r, i.e. v ≥ v₂ = √(2GM/r) = √2·v₁. At the Earth's surface v₂ ≈ 11.2 km/s. At v = v₂ the body escapes along a parabola, and its speed far away tends to zero. At v > v₂ the path is a hyperbola, and far from the planet the body keeps the speed v∞ = √(v² − v₂²). The total energy tells it all: E < 0 — the body is bound (ellipse), E = 0 — parabola, E > 0 — hyperbola.

Kepler's laws. Turn on “equal areas”: the line from the planet's centre to the satellite sweeps out equal areas in equal time intervals Δt (the second law). Near periapsis the sector is short and wide, near apoapsis it is long and narrow. This follows from conservation of angular momentum: r·v⊥ = const. The third law: T²/a³ = 4π²/GM is the same for every orbit around a given planet — compare this number in the Measurements tab for different shots.

How the simulator computes motion. The motion is not integrated step by step but computed exactly: the speed and height give the eccentricity and semi-major axis of the orbit, and the position at time t comes from Kepler's equation. So a circular orbit stays a perfect circle for any number of revolutions, and the time and point of landing are found exactly, not “on the nearest frame”.

Things to try. 1) Press “v₁” and fire — the ball returns to the mountain after one period T. 2) Slowly drag the speed from 7 to 11 km/s and watch the dashed line turn from an arc into a circle, an ellipse and a parabola. 3) Turn on equal areas at 9.5 km/s. 4) Raise the mountain to 2,000 km and check: the speed is below 7.9 km/s, yet the ball does not land. 5) Repeat the experiment on the Moon and on Jupiter. 6) In the “Test yourself” tab, calculate first, then check with a shot.

Orbits and Escape Velocities: Newton's Cannonball

Flight graphs

KEY FORMULAS

First cosmic (orbital) velocity

Second cosmic (escape) velocity

Energy and the shape of the orbit

Kepler's second law (law of equal areas)

Kepler's third law and the orbital period

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