An object has an initial velocity ';v0'; and an acceleration ';a';. How far (';x';) does it travel in a time t?

Please use algebra to explain and give me a solution. This is taken from the Idiots Guide to Physics. They say that the answer is x= v0 t +1/2 at squared. why?An object has an initial velocity ';v0'; and an acceleration ';a';. How far (';x';) does it travel in a time t?
If initial speed is V0 and the speed after a time t second is V,





Acceleration is change in speed per unit time.





Acceleration a = (V - VO) / t.





The average speed during this time t is





V (average) = (V + V0) / 2.





Note; if any quantity is changing uniformly, the average is found by adding the initial and final value and dividing the sum by 2.





The average speed can also be determined by dividing the total distance by total time of travel.





V (average) = S / t.





Therefore S/t = (V + V0) / 2.





S = (V + V0)* t / 2.





From a = (V - V0) / t we have V = V0 +at.





Therefore S = {V0 +at + VO} * t /2


--------------- = {2 V0 t + at t } /2





-------------- S = V0t + 0.5 a ttAn object has an initial velocity ';v0'; and an acceleration ';a';. How far (';x';) does it travel in a time t?
Initial velocity : v0


Acceleration : a


Initial distance : x0


Distance travelled : x


Time taken : t





v = dx /dt


dx = v dt


integrate both sides:


x 聽聽聽聽 聽螖t聽聽聽聽聽聽聽 螖t


鈭? dx = 鈭? v dt = 鈭? (v0 + a螖t )dt


x0聽聽聽 聽0聽聽聽聽聽聽聽 聽聽0





x 鈭?x0 = v0螖t + 陆 a螖t2





x = x0 + v0螖t + 陆 a螖t虏


as x0 is zero


and 螖t = t - 0 = t,


thus,





x = v0t + 陆 at虏





Have a nice day ...


:)
distance=(1/2)*a*t*t
u= initial velocity


v= final velocity





v= u + at


average speed S =[u + v]/2 = [u + u+ at]/2 = [2u+at]/2= u +i/2at








distance travelled x= average speed x time = S x t = [u+1/2at] t= ut + 1/2 at^2








substituting v0 for u as in ur problem,





x= v0 t +1/2 at^2
Take the equation v = u + at. By definition, this has to be correct.





then you'll get ds/dt = u + at


ds = (u+at).dt





ds = u.dt + at.dt





Integrate





s = ut + a * 1/2 t^2





s = ut + 1/2 a.t^2 - Voila!

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