# How do you get to v=d/t?

1. Apr 6, 2012

### dag45hol

Forgive the basic question but my Google Fu isn't strong enough in math.

I understand that for constant velocity the function of the distance relative to the time takes the form of: d=A*t+ B

Then of course the first derivative is the velocity: d'=v=A, which is a constant (expected since we started with a constant velocity).

Q1: how do you represent d' in Leibniz's notation?
Q2: how do you go from d' to the formula v=d/t?

Thanks

2. Apr 6, 2012

### I like Serena

Welcome to PF, dag45hol!

A1: The representation of d' in Leibniz's notation would be $d' = \frac {dd} {dt}$.
Of course this looks a bit ambiguous, so the letter d for distance is usually avoided when using Leibniz's notation.
Typically one of the letters s or x is used.

A2: The formula v=d/t is not always true in your case.
It will only hold if the B in your equation is zero.
In practice B is often chosen to be zero, so the formula v=d/t does hold.

Last edited: Apr 6, 2012
3. Apr 6, 2012

### HallsofIvy

Staff Emeritus
d= At+ B has a line as graph. For such a function, the derivative is a constant:
$$\lim_{h\to 0}\frac{(A(t+h)+ B)- (At+ B)}{h}= \lim_{h\to 0}\frac{At+ Ah+ B- At- b}{h}$$
$$= \lim_{h\to 0}\frac{Ah}{h}= A$$

If, with d(t)= At+ B, d(0)= B is NOT 0, v is NOT just "d/t". The average velocity, between t0 and t1, would be
$$\frac{d(t_1)- d(t_0)}{(t_1- t_0}= \frac{(At_1+ B)- (At_0+ B)}{t_1- t-0}= \frac{A(t_1- t_0)}{t_1- t_0}= A$$

As for Liebniz's notation, it would be dd/dt with the understanding, of course, that the second "d" is the distance function. There is no deep mathematics there but the possibility of confusion is one reason why it is more common to use "s" to represent the distance function.

4. Apr 6, 2012

### dag45hol

Lets replace d(istance) with s for clarity.

I get how ds/dt=v and how it comes to be a constant (A). What I don't get is how ds/dt=v=s/t

I get the feeling that this is obvious to you guys so please have patience.

Thanks

5. Apr 6, 2012

### Char. Limit

The thing you have to remember is that v=s/t is not always true. v is not always simply the quotient of distance and time. However, for constant velocity, the above equation holds. With non-constant velocity, it's better to use v=ds/dt.

6. Apr 6, 2012

### dag45hol

I understand that v=s/t is only good when v is a constant and that B must be zero for the function to hold.

The question is how do you go from v=ds/dt to the formula v=s/t?

Once again it seems that the answer is so obvious to you guys that you think I'm asking about other things.

ds/dt=v => fine
v=s/t => why? how do you go from ds/dt to s/t?

7. Apr 7, 2012

### I like Serena

You already had d'=A=v.

So your formula d=A*t+B, can also be written as d=v*t+B.
This can be manipulated into:

d=v*t+B
d-B=v*t
(d-B)/t=v
v=(d-B)/t

With B is 0, this becomes v=d/t.

8. Apr 7, 2012

### dag45hol

Thanks, I knew it had to be something obvious =)

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