{"id":257068,"date":"2022-09-10T11:46:52","date_gmt":"2022-09-10T06:16:52","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/th-displacement-x-of-a-particle-varies-with-time-t-as-x-ae-%ce%b1tbe%ce%b2t-where-a-b-%ce%b1-and-%ce%b2-are-positive-constants-the-velocity-of-the-particle-wi\/"},"modified":"2022-09-10T11:46:52","modified_gmt":"2022-09-10T06:16:52","slug":"th-displacement-x-of-a-particle-varies-with-time-t-as-x-ae-%ce%b1tbe%ce%b2t-where-a-b-%ce%b1-and-%ce%b2-are-positive-constants-the-velocity-of-the-particle-wi","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/physics\/th-displacement-x-of-a-particle-varies-with-time-t-asxae\/","title":{"rendered":"Th displacement x of a particle varies with time t as\u00a0x\u00a0=\u00a0ae-\u03b1t+be\u03b2t, where a, b,\u00a0\u03b1\u00a0and\u00a0\u03b2\u00a0are positive constants. The velocity of the particle will"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Th displacement x of a particle varies with time t as\u00a0x\u00a0=\u00a0ae-\u03b1t+be\u03b2t, where a, b,\u00a0\u03b1\u00a0and\u00a0\u03b2\u00a0are positive constants. The velocity of the particle will","custom_permalink":"question\/physics\/th-displacement-x-of-a-particle-varies-with-time-t-asxae\/"},"categories":[],"acf":{"question":"<p>Th displacement x of a particle varies with time t as&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mrow><mi mathvariant=\"normal\">x<\/mi><mo>&nbsp;<\/mo><mo>=<\/mo><mo>&nbsp;<\/mo><msup><mi>ae<\/mi><mrow><mo>-<\/mo><mi>\u03b1t<\/mi><\/mrow><\/msup><mo>+<\/mo><msup><mi>be<\/mi><mi>\u03b2t<\/mi><\/msup><\/mrow><\/mstyle><\/math>, where a, b,&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mrow><mi mathvariant=\"normal\">\u03b1<\/mi><mo>&nbsp;<\/mo><mi>and<\/mi><mo>&nbsp;<\/mo><mi mathvariant=\"normal\">\u03b2<\/mi><\/mrow><\/mstyle><\/math>&nbsp;are positive constants. The velocity of the particle will<\/p>","options":[{"option":"<p>go on decreasing with time<\/p>","correct":false},{"option":"<p>be independent of&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mrow><mi mathvariant=\"normal\">\u03b1<\/mi><mo>&nbsp;<\/mo><mi>and<\/mi><mo>&nbsp;<\/mo><mi mathvariant=\"normal\">\u03b2<\/mi><\/mrow><\/mstyle><\/math><\/p>","correct":false},{"option":"<p>drop to zero when&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mrow><mi mathvariant=\"normal\">\u03b1<\/mi><mo>&nbsp;<\/mo><mo>=<\/mo><mo>&nbsp;<\/mo><mi mathvariant=\"normal\">\u03b2<\/mi><\/mrow><\/mstyle><\/math><\/p>","correct":false},{"option":"<p>go on increasing with time<\/p>","correct":true}],"solution":"<p>We are given<\/p><p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mi mathvariant=\"normal\">x<\/mi><mo>&nbsp;<\/mo><mo>=<\/mo><mo>&nbsp;<\/mo><msup><mi>ae<\/mi><mrow><mo>-<\/mo><mi>\u03b1<\/mi><mi>t<\/mi><\/mrow><\/msup><mo>+<\/mo><msup><mi>be<\/mi><mi>\u03b2t<\/mi><\/msup><\/mstyle><\/math><\/p><p>Velocity &nbsp; v =&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mrow><mfrac><mi>dx<\/mi><mi>dt<\/mi><\/mfrac><mo>&nbsp;<\/mo><mo>=<\/mo><mo>&nbsp;<\/mo><mfrac><mi mathvariant=\"normal\">d<\/mi><mi>dt<\/mi><\/mfrac><mo>(<\/mo><msup><mi>ae<\/mi><mrow><mo>-<\/mo><mi>\u03b1t<\/mi><\/mrow><\/msup><mo>+<\/mo><msup><mi>be<\/mi><mi>\u03b2t<\/mi><\/msup><mo>)<\/mo><\/mrow><\/mstyle><\/math><\/p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mrow><mi mathvariant=\"normal\">a<\/mi><mo>.<\/mo><msup><mi mathvariant=\"normal\">e<\/mi><mrow><mo>-<\/mo><mi>\u03b1t<\/mi><\/mrow><\/msup><mo>(<\/mo><mo>-<\/mo><mi mathvariant=\"normal\">\u03b1<\/mi><mo>)<\/mo><mo>+<\/mo><msup><mi>be<\/mi><mi>\u03b2t<\/mi><\/msup><mo>.<\/mo><mi mathvariant=\"normal\">\u03b2<\/mi><\/mrow><\/mstyle><\/math><\/p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; =&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mrow><mo>-<\/mo><msup><mi>a\u03b1e<\/mi><mrow><mo>-<\/mo><mi>\u03b1t<\/mi><\/mrow><\/msup><mo>+<\/mo><msup><mi>b\u03b2e<\/mi><mi>\u03b2t<\/mi><\/msup><\/mrow><\/mstyle><\/math><\/p><p>Acceleration =&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mrow><mo>-<\/mo><msup><mi>a\u03b1e<\/mi><mrow><mo>-<\/mo><mi>\u03b1t<\/mi><\/mrow><\/msup><mo>(<\/mo><mo>-<\/mo><mi mathvariant=\"normal\">\u03b1<\/mi><mo>)<\/mo><mo>&nbsp;<\/mo><mo>+<\/mo><msup><mi>b\u03b2e<\/mi><mi>\u03b2t<\/mi><\/msup><mo>.<\/mo><mi mathvariant=\"normal\">\u03b2<\/mi><\/mrow><\/mstyle><\/math><\/p><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; =&nbsp;<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mstyle mathsize=\"14px\"><mrow><msup><mi>a\u03b1<\/mi><mn>2<\/mn><\/msup><msup><mi mathvariant=\"normal\">e<\/mi><mrow><mo>-<\/mo><mi>\u03b1t<\/mi><\/mrow><\/msup><mo>+<\/mo><msup><mi>b\u03b2<\/mi><mn>2<\/mn><\/msup><msup><mi mathvariant=\"normal\">e<\/mi><mi>\u03b2t<\/mi><\/msup><\/mrow><\/mstyle><\/math><\/p><p>Acceleration is positive so velocity goes on increasing with time<\/p>","subject":""},"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Th displacement x of a particle varies with time t as\u00a0x\u00a0=\u00a0ae-\u03b1t+be\u03b2t, where a, b,\u00a0\u03b1\u00a0and\u00a0\u03b2\u00a0are positive constants. 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