Physics · Gravitation and the universe · Grade 10

Newton's cannon: an artificial satellite

At what speed must an object be launched so that it never falls back to Earth?

The situation

This is how Isaac Newton pictured it: put a cannon on top of a tall mountain and fire it horizontally. A ball fired slowly lands nearby; a faster one lands farther away. If the speed is great enough, the surface of the Earth curves away beneath the ball and it never reaches the ground — it becomes a satellite. This is exactly how the ISS, GPS and television satellites fly today.

Topic

Universal gravitation, cosmic velocities. Newton's cannon, the ISS and a geostationary satellite

Key ideas

Law of universal gravitation
Cosmic velocities
Kepler's laws

Formulas

F = G \frac{M m}{r^{2}}

The law of universal gravitation: the force is inversely proportional to the square of the distance.

v_{1} = \sqrt{\frac{G M}{r}}

First cosmic (circular-orbit) speed: gravity provides the centripetal acceleration, $G M / r^{2} = v^{2} / r$ .

v_{2} = \sqrt{\frac{2 G M}{r}} = \sqrt{2} v_{1}

Second cosmic (escape) speed: the total energy is zero.

E = \frac{v^{2}}{2} - \frac{G M}{r}, e = \sqrt{1 + \frac{2 E L^{2}}{{(G M)}^{2}}}

The specific energy $E$ and the angular momentum $L = | \vec{r} \times \vec{v} |$ determine the shape of the orbit (eccentricity $e$ ).

T = 2 π \sqrt{\frac{a^{3}}{G M}}

Kepler's third law: the period depends on the semi-major axis.

Theory

A satellite is always falling

The only force acting on a body in orbit is gravity — it is in free fall. But its horizontal speed is so great that the distance it falls towards the Earth in one second (≈ 4.9 m) equals the drop of the Earth's curving surface. That is why the astronauts on the ISS are weightless: they are falling together with the satellite.

Energy determines the shape of the orbit

$E < 0$ — a closed ellipse (a circle is a special case). $E = 0$ — a parabola, $E > 0$ — a hyperbola: the body leaves the Earth for good. If you raise the speed above $v_{1}$ , the ellipse stretches out; when it reaches $v_{2}$ , the orbit “opens”.

Kepler's second law

A satellite speeds up as it comes close to the Earth and slows down as it moves away: the radius vector sweeps out equal areas in equal times. This follows from the conservation of angular momentum $L = r v_{⟂}$ . Switch on the “Kepler's second law” button and compare the sectors.

The third cosmic speed and spacecraft

At the Earth's orbit (1 AU) the speed needed to escape the Sun's gravity is $\sqrt{2} \cdot 29.8 \approx 42.1$ km/s. The Earth itself moves at 29.8 km/s, so a spacecraft launched in the direction of the Earth's motion only needs to keep a speed of $v_{\infty} \approx 12.3$ km/s after leaving the Earth's gravity. From conservation of energy, $v_{3} = \sqrt{v_{\infty}^{2} + v_{2}^{2}} \approx 16.65$ km/s — the third cosmic speed (from the Earth's surface, ignoring the Earth's rotation and air resistance). When the object escapes from the Earth, the simulation shows the “speed at infinity” $v_{\infty}$ — compare it with 12.3 km/s. Interplanetary spacecraft run their engines only briefly and then coast freely around the Sun along a conic section; gaining speed by passing close to a planet is a gravity assist: Voyager 1 and Voyager 2 flew past Jupiter and Saturn and reached a speed that takes them out of the Solar System.

Geostationary orbit

If the period of revolution equals the period of the Earth's rotation about its axis (a sidereal day — 23 h 56 min 4 s), a satellite above the equator appears to “freeze” at one point in the sky as seen from the Earth. The radius of this orbit is $r = \sqrt[3]{G M T^{2} / 4 π^{2}} \approx 42 164$ km, that is, an altitude of 35 786 km above the surface.

Did you know?

The ISS flies at about 7.66 km/s and circles the Earth once in about 92 minutes — the crew sees the Sun rise 15–16 times a day.
For the Earth's surface the first cosmic speed is ≈ 7.9 km/s and the second (escape) speed ≈ 11.2 km/s. Newton's “cannon on a mountain” thought experiment is illustrated with a diagram in his book “Principia” (“The System of the World”).

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

The Solar System — Grade 7
Scales of the universe — Grade 9
Special relativity — Grade 10
Law of universal gravitation — Grade 10
Relativistic dynamics — Grade 10
Kepler's laws — Grade 10
Physics of stars — Year 1
The expanding universe — Year 1
Black holes — Year 4