A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite = 200 kg; mass of the earth = 6.0 ×1024 kg; radius of the earth = 6.4 ×106 m; G = 6.67 × 10–11 N m2 kg–2. | Learn NCERT solution | Education portal Class 11 Physics | Study online Unit-8 Gravitation

Q.19:- A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite = 200 kg; mass of the earth = 6.0 ×1024 kg; radius of the earth = 6.4 ×106 m; G = 6.67 × 10–11 N m2 kg–2.

Mass of the Earth, M = 6.0 × 1024 kg
Mass of the satellite, m = 200 kg
Radius of the Earth, Re = 6.4 × 106 m
Universal gravitational constant, G = 6.67 × 10–11 Nm2kg–2
Height of the satellite, h = 400 km = 4 × 105 m = 0.4 ×106 m
Total energy of the satellite at height h = (1/2)mv2 + [ -GMem / (Re + h) ]
Orbital velocity of the satellite, v = [ GMe / (Re + h) ]1/2
Total energy of height, h = (1/2)GMem / (Re + h) – GMem / (Re + h)  =  -(1/2)GMem / (Re + h)
The negative sign indicates that the satellite is bound to the Earth. This is called bound energy of the satellite.
Energy required to send the satellite out of its orbit = – (Bound energy)
= (1/2) GMem / (Re + h)
= (1/2) × 6.67 × 10-11 × 6 × 1024 × 200 / (6.4 × 106 + 0.4 × 106)
= 5.9 × 109 J.
{"ft_lang_label":"__label__en","ft_lang_prob":0.8135466,"math_prob":0.9991986,"size":1057,"snap":"2019-43-2019-47","text_gpt3_token_len":384,"char_repetition_ratio":0.20322888,"word_repetition_ratio":0.025751073,"special_character_ratio":0.44276252,"punctuation_ratio":0.12033195,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9992886,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-22T21:49:35Z\",\"WARC-Record-ID\":\"<urn:uuid:4b7d536c-07b9-4d3c-bd01-ec7c606388db>\",\"Content-Length\":\"40075\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0c8bebf3-caac-48a8-ac86-692eaa02a44c>\",\"WARC-Concurrent-To\":\"<urn:uuid:b21c659f-2d69-40fd-9bf6-6397ec571158>\",\"WARC-IP-Address\":\"142.93.56.99\",\"WARC-Target-URI\":\"http://www.tutionteacher.in/a-satellite-orbits-the-earth-at-a-height-of-400-km-above-the-surface-how-much-energy-must-be-expended-to-rocket-the-satellite-out-of-the-earths-gravitational-influence-mass-of-the-satellite/\",\"WARC-Payload-Digest\":\"sha1:42GETUC6W6DSLAF6P2NU7AUDGU7CB2EZ\",\"WARC-Block-Digest\":\"sha1:UTRGZIJQYXFJIZWUVM5E24LC2X3UWUZY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987824701.89_warc_CC-MAIN-20191022205851-20191022233351-00111.warc.gz\"}"}