TE Enriched Sample (E)

Orbital Motions under Gravity 3 74 B Conservation of mechanical energy ( MC: DSE 2012 Q1.2, 2016 Q1.2, 2018 Q1.1)( SQ: DSE 2020 Q1ab) For orbital motion in space , mechanical energy is still conserved (Fig. 3.18), but the general formula for PE, U = − GMm / r , should be used. For any two positions 1 and 2 in the orbit, Fig. 3.17 Conservation of mechanical energy near the surface of the Earth cannon cannon ball PE = mgh 1 KE = mv 1 2 PE = mgh 2 KE = mv 2 2 1 2 1 2 h 1 ground h 2 position 2 position 1 Fig. 3.18 Conservation of mechanical energy for orbital motion in space r 2 r 1 spacecraft (mass m ) v 2 v 1 Earth (mass M ) GMm PE = − KE = mv 1 2 KE = mv 2 2 1 2 1 2 r 2 GMm PE = − r 1 position 2 position 1 mv r GMm mv r GMm 2 1 2 1 1 2 1 2 2 2 - = - This equation applies to any orbital motion of celestial bodies or spacecraft under a gravitational force. When a spacecraft is propelled or decelerated by firing its rockets, however, mechanical energy is not conserved and the above equation does not apply. In Force and Motion , we have learnt that mechanical energy is equal to the sum of kinetic energy (KE) and gravitational PE. Consider an object of mass m projected near the surface of the Earth (Fig. 3.17). When it moves from position 1 to 2, conservation of mechanical energy gives KE PE KE PE mv mgh mv mgh 2 1 2 1 1 1 2 2 1 2 1 2 2 2 + = + + = + Its orbit will be changed. See Example 3.7 on p. 77. Simulation Energy in orbital motion Teaching notes Motion of a satellite with energy loss is not included in the HKDSE syllabus. Teaching notes Remind Ss that, in Fig 3.18, • the spacecraft has lowest PE and highest KE at position 1, and highest PE and lowest PE at position 2. • the magnitude of the acceleration of the spacecraft is the greatest when it is moving the fastest (position 1). At that point, it is nearest to the Earth, so the force of gravity is the greatest. Over 50% students fail to realise this. (Ref: DSE 2018 Q1.1) 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.