(a) Assuming that the earth (mass ME ) orbits the sun (mass MS ) in a circle of radius R and with a speed v, write down the equation of motion for the earth. Hence show that G MS = v2 R (b) A comet is in orbit around the sun in the same plane as the earth's orbit, as shown in Fig. 5.8. Its distance of closest approach to the sun's centre is R/2, at which point it has speed 2v. Using the condition for the Earth's orbit given in (a), show that the comet's total energy is zero. (Neglect the effect of the earth on the comet.) (c) Use conservation of angular momentum to determine the component of the comet's velocity which is tangential to the earth's orbit at the point P, where the comet's orbit crosses that of the earth. (d) Use conservation of energy to find its speed at the point P. Hence show that the comet crosses the earth's orbit at an angle of 45°.
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