TE Enriched Sample (E)

Energy in orbital motion Checkpoint 5 75 3.3 The perihelion and aphelion distances of Mars are 1.38 AU and 1.67 AU, respectively. The orbital speed of Mars at the aphelion is 21 900 m s −1 . Find the orbital speed of Mars at the perihelion. Given the mass of the Sun = 1.99 × 10 30 kg and 1 AU = 1.5 × 10 11 m. Take G = 6.67 × 10 −11 N m 2 kg −2 . Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The perihelion and aphelion distances expressed in meters are r 1 = 1.38 × (1.5 × 10 11 ) = 2.07 × 10 11 m r 2 = 1.67 × (1.5 × 10 11 ) = 2.505 × 10 11 m By conservation of mechanical energy, ( . ) ( . ) . . . m s mv r GMm mv r GMm v v GM r r v 2 1 2 1 2 1 2 1 1 1 2 1 21900 6 67 10 1 99 10 2 07 10 1 2 505 10 1 3 511 10 26 501 1 2 1 2 2 2 1 2 2 2 1 2 2 11 30 11 11 8 1 1 # # # # # # - = - = + - = + - = = - - J L K J L K N P OO N P OO The orbital speed of Mars at perihelion is 26 500 m s −1 . To calculate the gravitational PE, the distances must be expressed in SI units, i.e. metres. The orbital speed of Mars Example 3.6 Mars aphelion perihelion Sun r 2 r 1 v 2 v 1 1. A planet moves around a star in an elliptical orbit as shown. Among positions A , B and C , at which does the planet have the largest (a) gravitational PE? (b) kinetic energy? (c) mechanical energy? Explain briefly. A B C star planet 2. An artificial satellite orbits around the Earth in an elliptical orbit. Its maximum and minimum distances from the centre of the Earth are 9370 km and 7370 km, respectively. If the speed of the satellite is 7780 m s −1 when it is closest to the Earth, find its speed when it is farthest from the Earth. The mass of the Earth is 5.97 × 10 24 kg. Since mechanical energy is conserved, mv r GMm mv r GMm v v 2 1 2 1 2 1 1 2 1 2 2 2 1 2 1 - = - = = Teaching notes The orbit of Mars is more elliptical than those of other planets. Teaching notes You may also use the conservation of angular momentum for an easier calculation. However, it is not required in the ‘Astronomy and Space Science’ syllabus. v GM r r 2 1 1 1 2 2 1 2 + - d n 6120 m s −1 A , B , C A C Sample © United Prime Educational Publishing (HK) Limited, Pearson Education Asia Limited 2023 All rights reserved; no part of this publication may be reproduced, photocopied, recorded or otherwise, without the prior written permission of the Publishers.