Motion in a gravitational field
The faster a body is thrown horizontally near the Earth, the farther it lands; at a high enough speed its “fall” matches the curving of the Earth’s surface and the body goes round the Earth along a circle – an artificial satellite. For a satellite gravity gives the centripetal acceleration: mg = mυ²/R, so the speed in an orbit near the surface is υ₁ = √(gR) ≈ 7.9 km/s (R ≈ 6400 km, g ≈ 9.8 m/s²); this is the first cosmic velocity. At this speed the satellite circles the Earth in ≈ 84 min. In an orbit at height h the speed υ = √(gR²/(R + h)) is lower and the period longer. The geostationary orbit, with a period of 24 h, is about 42 000 km from the Earth’s centre (≈ 36 000 km above the surface) in the equatorial plane, and the satellite appears to hang motionless over one point. To leave the Earth completely a speed of the second cosmic velocity ≈ 11.2 km/s is needed, and to leave the Solar System the third, ≈ 16.7 km/s. The first artificial satellite was launched on 4 October 1957, the first person flew into space on 12 April 1961, and people landed on the Moon on 20 July 1969.
In class: make a simple “Earth–satellite” model with cardboard and string (use scissors only with the teacher’s help and carefully). Whirl a ball on a string slowly above the “Earth” and feel how the tension grows as the speed rises (speed determines the centripetal force). Keep everyone clear while whirling.