# How Does Calculus Explain How V=S/T & DV=DS/DT?

• abrahamjp
In summary, Galileo's equations for velocity and displacement under constant acceleration, v=u+at and s=ut+1/2at^2, were based on the concept of average velocity and acceleration. However, with the use of calculus and analytic geometry, we are able to find the instantaneous velocity and acceleration, which are represented by dv/dt and v(t), respectively. This allows for a more accurate and precise understanding of motion under varying accelerations.

#### abrahamjp

Dear Sirs,

I am wondering what is the difference between v=s/t & dv = ds/dt where v-velocity,s-displacement,t-time.

Consider Gallelio's equations->
v=u+at--(equation-1) &
s=ut+1/2at^2--(equation-2)
where u-initial velocity & a-acceleration

My doubt is on following point;

If we do,v = s/t in equation-2,we get->v=u+1/2at not v=u+at
but if we do,dv=ds/dt in equation-2,we get=>v=u+at ,exactly the equation we want.

why only differentiation give the result,not just mere divison?

v=s/t is the AVERAGE velocity over the total time INTERVAL "t".
Thus, it does NOT give the velocity at the INSTANT "t"

abrahamjp said:
I am wondering what is the difference between v=s/t & dv = ds/dt where v-velocity,s-displacement,t-time.

Hi !
dv = ds/dt has no meaning at all !
dv is an infinitesimal value (a very small variation of v). Do not confuse it with v.
ds is a small displacement during dt a small variation of time.
So, ds/dt is the speed at time t, which is not infinitesimal, hense not equal to dv.
Do not confuse the average speed s/t with the instantaneous speed ds/dt. Use two different symbols, not v for both.
dt is the small variation of speed during a small variation of time. So, dv/dt is the acceleration.

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dv/dt is at an instantaneous time... Not division.

iRaid said:
dv/dt is at an instantaneous time... Not division.
It's the acceleration at an arbitrary time.

Galileo's equations work only for constant accelerations; the theory was worked out by the "Oxford Calculators":
http://en.wikipedia.org/wiki/Oxford_Calculators

Galilleo's experiments showed that gravity was constant (at the surface of the earth), and was thus able to apply these equations.

Calculus allows you to work with arbitrary accelerations; instead of working with algebraic averages it makes use of limits. Between the time of Galileo and Newton, Rene Descartes invented analytic geometry ... this is the tool required to move from the geometric analysis of Galileo to the calculus of Newton and Leibniz.

equation $v=u+at$ is merely rearranged from the average acceleration formula $\displaystyle\frac{v-u}{t}=a$.

In calculus we get the instantaneous versions of these, $\displaystyle\frac{dv}{dt} = a$, and $v(t)=\displaystyle\int a(t) dt$