An object has an initial velocity "v0" and an acceleration "a". How far ("x") does it travel in a time t?

Question

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?

Answer 1

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 tt

Answer 2

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

Answer 3

Initial velocity : v0
Acceleration : a
Initial distance : x0
Distance travelled : x
Time taken : t

v = dx /dt
dx = v dt
integrate both sides:
x       Δt        Δt
∫ dx = ∫ v dt = ∫ (v0 + aΔt )dt
x0     0          0

x − x0 = v0Δt + ½ aΔt2

x = x0 + v0Δt + ½ aΔt²
as x0 is zero
and Δt = t - 0 = t,
thus,

x = v0t + ½ at²

Have a nice day ...
:)

Answer 4

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!

Answer 5

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Answer 6

distance=(1/2)*a*t*t