Question

Can somebody exlplain to me the meaning of this equation?

Answer 1

\[V(t)=V _{0}+\int\limits_{x _{0}}^{x }a(t)dt\]

Answer 2

do you know about integral calculus?

Answer 3

Um.. just a little bit

Answer 4

OK
Do you know about the kinetic equations of motion v=u+at etc.?

Answer 5

yeah

Answer 6

The equations you see here
v=u+at
s=ut +0.5 at^2

only apply for CONSTANT acceleration.

The equation you have written is related to those equations, but it takes into account that acceleration may not be constant.

This is very often the case - e.g. a rocket burns fuel, and hence the mass reduces in time, but the thrust remains the same - so so the acceleration increases.

2 bodies attracted by gravity have an INCREASING force between them as they get closer - so the acceleration increases

Answer 7

your equation say the Velocity as a function of time is the integral (wrt time) of the acceleration (as a function of time)
After each integration you must remember to add the 'constant of integration'
In your equation this constant is V0, the initial velocity.

I can say more if you wish - or does that cover your question?

Answer 8

IF a(t) = a in your equation, (i.e. a is constant) THEN the more familiar equations can be derived from your equation as shown below:

a=dv/dt

dv=adt

∫dv=∫adt

v=at+c

but when t = 0 v = vi
Therefore v = vi + at

But v = ds/dt

So ds/dt = vi + at (at this point vi is a constant)

so
s=∫(vi+at)dt

s=vit+12at2+c


now - it is usual ( but not necessary) that s = 0 when t = 0 so in this case c=0
but in general
s=s0+vit+12at2