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The-position-of-a-body-at-time-t-in-seconds-is-given-by-s-t-3-4-what-is-the-velocity-and-acceleration-of-the-object-at-time-t-5-Mastermind-




Question Number 170411 by Mastermind last updated on 23/May/22
The position of a body at time t (in  seconds) is given by s=t^3 −4, what is  the velocity and acceleration of the  object at time t=5 ?    Mastermind
$${The}\:{position}\:{of}\:{a}\:{body}\:{at}\:{time}\:{t}\:\left({in}\right. \\ $$$$\left.{seconds}\right)\:{is}\:{given}\:{by}\:{s}={t}^{\mathrm{3}} −\mathrm{4},\:{what}\:{is} \\ $$$${the}\:{velocity}\:{and}\:{acceleration}\:{of}\:{the} \\ $$$${object}\:{at}\:{time}\:{t}=\mathrm{5}\:? \\ $$$$ \\ $$$${Mastermind} \\ $$
Answered by alephzero last updated on 23/May/22
velocity = (ds/dt)  acceleration = (d^2 s/dt^2 )  (ds/dt) = ((d(t^3 −4))/dt) = ((d(t^3 ))/dt)−((d(4))/dt) =  = 3t^2 −0 = 3t^2   (d^2 s/dt^2 ) = ((d((ds/dt)))/dt) = ((d(3t^2 ))/dt) = 3((d(t^2 ))/dt) = 3×2t  = 6t  velocity at time 5 = (ds/dt)∣_5  = 3(5)^2  =  = 3×25 = 75  acceleration at time 5 = (d^2 s/dt^2 )∣_5  = 6×5  = 30
$$\mathrm{velocity}\:=\:\frac{{ds}}{{dt}} \\ $$$$\mathrm{acceleration}\:=\:\frac{{d}^{\mathrm{2}} {s}}{{dt}^{\mathrm{2}} } \\ $$$$\frac{{ds}}{{dt}}\:=\:\frac{{d}\left({t}^{\mathrm{3}} −\mathrm{4}\right)}{{dt}}\:=\:\frac{{d}\left({t}^{\mathrm{3}} \right)}{{dt}}−\frac{{d}\left(\mathrm{4}\right)}{{dt}}\:= \\ $$$$=\:\mathrm{3}{t}^{\mathrm{2}} −\mathrm{0}\:=\:\mathrm{3}{t}^{\mathrm{2}} \\ $$$$\frac{{d}^{\mathrm{2}} {s}}{{dt}^{\mathrm{2}} }\:=\:\frac{{d}\left(\frac{{ds}}{{dt}}\right)}{{dt}}\:=\:\frac{{d}\left(\mathrm{3}{t}^{\mathrm{2}} \right)}{{dt}}\:=\:\mathrm{3}\frac{{d}\left({t}^{\mathrm{2}} \right)}{{dt}}\:=\:\mathrm{3}×\mathrm{2}{t} \\ $$$$=\:\mathrm{6}{t} \\ $$$$\mathrm{velocity}\:\mathrm{at}\:\mathrm{time}\:\mathrm{5}\:=\:\frac{{ds}}{{dt}}\mid_{\mathrm{5}} \:=\:\mathrm{3}\left(\mathrm{5}\right)^{\mathrm{2}} \:= \\ $$$$=\:\mathrm{3}×\mathrm{25}\:=\:\mathrm{75} \\ $$$$\mathrm{acceleration}\:\mathrm{at}\:\mathrm{time}\:\mathrm{5}\:=\:\frac{{d}^{\mathrm{2}} {s}}{{dt}^{\mathrm{2}} }\mid_{\mathrm{5}} \:=\:\mathrm{6}×\mathrm{5} \\ $$$$=\:\mathrm{30} \\ $$
Commented by Mastermind last updated on 24/May/22
Thanks
$${Thanks} \\ $$

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