{"id":542045,"date":"2023-04-23T05:39:30","date_gmt":"2023-04-23T00:09:30","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/an-artificial-satellite-is-moving-in-a-circular-orbit-around-the-earth-with-a-speed-equal-to-half-the-magnitude-of-escape-velocity-from-the-earth\/"},"modified":"2025-06-20T15:17:13","modified_gmt":"2025-06-20T09:47:13","slug":"an-artificial-satellite-is-moving-in-a-circular-orbit-around-the-earth-with-a-speed-equal-to-half-the-magnitude-of-escape-velocity-from-the-earth","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/physics\/an-artificial-satellite-is-moving-in-a-circular-orbit-around\/","title":{"rendered":"An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth."},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth.","custom_permalink":"question\/physics\/an-artificial-satellite-is-moving-in-a-circular-orbit-around\/"},"categories":[],"acf":{"question":"<p>An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth.<\/p>","options":null,"solution":"The escape velocity from Earth (<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>v<\/mi><mi>e<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">v_e<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">e<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>) is the speed needed to break free from Earth's gravitational pull without further propulsion. The formula for escape velocity is:\r\n\r\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>v<\/mi><mi>e<\/mi><\/msub><mo>=<\/mo><msqrt><mrow><mn>2<\/mn><mi>g<\/mi><mi>R<\/mi><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">v_e = \\sqrt{2gR}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">e<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">2<span class=\"mord mathnormal\">g<\/span><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nWhere:\r\n<ul>\r\n \t<li><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>g<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">g<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">g<\/span><\/span><\/span><\/span> is the acceleration due to Earth's gravity (approximately <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>9.8<\/mn><mtext>\u2009<\/mtext><msup><mtext>m\/s<\/mtext><mn>2<\/mn><\/msup><\/mrow><annotation encoding=\"application\/x-tex\">9.8 \\, \\text{m\/s}^2<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">9.8<\/span><span class=\"mord\"><span class=\"mord text\">m\/s<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>),<\/li>\r\n \t<li><span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>R<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">R<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span> is the radius of the Earth (approximately <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>6<\/mn><mo separator=\"true\">,<\/mo><mn>371<\/mn><mtext>\u2009<\/mtext><mtext>km<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">6,371 \\, \\text{km}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">6<\/span><span class=\"mpunct\">,<\/span><span class=\"mord\">371<\/span><span class=\"mord text\"><span class=\"mord\">km<\/span><\/span><\/span><\/span><\/span> or <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>6.371<\/mn><mo>\u00d7<\/mo><mn>1<\/mn><msup><mn>0<\/mn><mn>6<\/mn><\/msup><mtext>\u2009<\/mtext><mtext>m<\/mtext><\/mrow><annotation encoding=\"application\/x-tex\">6.371 \\times 10^6 \\, \\text{m}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\">6.371<\/span><span class=\"mbin\">\u00d7<\/span><\/span><span class=\"base\"><span class=\"mord\">1<\/span><span class=\"mord\">0<span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord text\"><span class=\"mord\">m<\/span><\/span><\/span><\/span><\/span>).<\/li>\r\n<\/ul>\r\nThe satellite is moving at half the escape velocity, so its speed <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>v<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">v<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">v<\/span><\/span><\/span><\/span> is:\r\n\r\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>v<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msub><mi>v<\/mi><mi>e<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">v = \\frac{1}{2} v_e<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">v<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">21<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord mathnormal\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">e<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nSubstituting the expression for <span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>v<\/mi><mi>e<\/mi><\/msub><\/mrow><annotation encoding=\"application\/x-tex\">v_e<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathnormal\">v<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">e<\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>:\r\n\r\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>v<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><msqrt><mrow><mn>2<\/mn><mi>g<\/mi><mi>R<\/mi><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">v = \\frac{1}{2} \\sqrt{2gR}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">v<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord\"><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\">21<\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\">2<span class=\"mord mathnormal\">g<\/span><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\r\n\r\nSimplifying:\r\n\r\n<span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><math display=\"block\" xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>v<\/mi><mo>=<\/mo><msqrt><mrow><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>g<\/mi><mi>R<\/mi><\/mrow><\/msqrt><\/mrow><annotation encoding=\"application\/x-tex\">v = \\sqrt{\\frac{1}{2} gR}<\/annotation><\/semantics><\/math><\/span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mord mathnormal\">v<\/span><span class=\"mrel\">=<\/span><\/span><span class=\"base\"><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"svg-align\"><span class=\"mord\"><span class=\"mfrac\">21<span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"mord mathnormal\">g<\/span><span class=\"mord mathnormal\">R<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","subject":"Physics"},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth. - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"An artificial satellite is moving in a circular orbit around the earth with a speed equal to half the magnitude of escape velocity from the earth.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/question\/physics\/an-artificial-satellite-is-moving-in-a-circular-orbit-around\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" 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