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

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.
  • #1
abrahamjp
6
0
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?
 
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  • #2
v=s/t is the AVERAGE velocity over the total time INTERVAL "t".
Thus, it does NOT give the velocity at the INSTANT "t"
 
  • #3
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.
 
Last edited:
  • #4
dv/dt is at an instantaneous time... Not division.
 
  • #5
iRaid said:
dv/dt is at an instantaneous time... Not division.
It's the acceleration at an arbitrary time.
 
  • #6
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.
 
  • #7
equation [itex]v=u+at[/itex] is merely rearranged from the average acceleration formula [itex]\displaystyle\frac{v-u}{t}=a[/itex].

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

1. How does calculus explain the relationship between velocity (V) and displacement (S) over time (T)?

In calculus, velocity is defined as the rate of change of displacement with respect to time. This can be represented mathematically as V = dS/dT, where dS is the change in displacement and dT is the change in time. This means that the slope of the curve representing displacement over time is equal to the velocity at any given point.

2. What is the significance of the derivative in the equation DV = dS/dT?

The derivative in this equation represents the instantaneous rate of change of velocity with respect to time. It allows us to calculate the acceleration of an object at any given moment, as acceleration is defined as the change in velocity over time.

3. How does calculus explain the concept of average velocity?

In calculus, average velocity is calculated by taking the total change in displacement and dividing it by the total change in time. This can also be represented graphically as the slope of a straight line connecting two points on a displacement-time graph.

4. Can calculus be used to model and predict the motion of objects?

Yes, calculus is an essential tool in modeling and predicting the motion of objects. By using the principles of differentiation and integration, we can determine the position, velocity, and acceleration of an object at any given time, as well as make predictions about its future motion.

5. How does calculus explain the concept of instantaneous velocity?

In calculus, instantaneous velocity is the velocity of an object at a specific point in time, or at a specific instant. This is found by taking the derivative of the displacement-time function at that point. It gives us the exact velocity of the object at that moment, rather than an average over a given period of time.

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