A particle is moving along a straight line according to the given equation of motion, where s ft is the directed distance of the particle from the origin at t sec. Find the time when the instantaneous acceleration is zero, and then find the directed distance of the particle from the
a. π = (πππ)/ (πππ+ππ) β 2/5ππ , tβ₯π
b. π= πππ + π βππ+π, tβ₯π
a)
"s=\\frac{125}{16t+32}-\\frac{2}{5t^5},t\\ge0"
Acceleration ="\\frac{d^2s}{dt^2}"
"\\frac{ds}{dt}(s)=\\frac{2}{t^6}-\\frac{2000}{(16t+32)^2}"
"\\frac{d^2s}{dt^2}(s)=\\frac{64000}{(16t+32)^3}-\\frac{12}{t^7}"
Time when Instantaneous acceleration is zero,
"\\frac{64000}{(16t+32)^3}-\\frac{12}{t^7}=0"
"64000t^7-12(16t+32)^3=0"
"64000t^7-12(4096t^3+24576t^2+49152t+32768)=0"
Solving to get "t\\approx1.6839"
Time=1.68 seconds
"s=\\frac{125}{16t+32}-\\frac{2}{5t^5}"
"=\\frac{125}{16(1.68)+32}-\\frac{2}{5(1.68)^5}"
=0.055 ft
b)
"s=9t^2+2\\sqrt{2t}+1,t\\ge0"
"\\frac{ds}{dt}=18t+\\frac{\\sqrt{2}}{\\sqrt{t}}"
"\\frac{d^2s}{dt^2}=18-\\frac{1}{\\sqrt{2}t^{\\frac{3}{2}}}"
"18-\\frac{1}{\\sqrt{2}t^{\\frac{3}{2}}}=0"
"18({\\sqrt{2}t^{\\frac{3}{2}}})-1=0"
"t^{\\frac{3}{2}}=\\frac{1}{25.46}"
"t=0.12seconds"
"s=9t^2+2\\sqrt{2t}+1"
"=9(0.12)^2+2\\sqrt{2(0.12)}+1"
=0.1296+1.9798
=2.1094 ft
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