[music].
Let's suppose that I have a formula for my
velocity.
For instance I made my velocity at time t
is something like 3 minus 10 times t.
With that equation in hand, can I then
determine my position?
Right?
Velocity and position are related in some
way.
The derivative of my position is, well
let's, let's write that fact down.
P of t is my position at time t.
V of t is my velocity at time t.
And the crucial thing here is if the
derivative of my position, right?
The rate of change of my position is
velocity.
So, the derivative of p is v.
So, I've written that as a derivative
statement but I could also write it as an
antiderivative statement.
So, I can write that p of t is an
antiderivative of v of t, right?
Because p differentiates to v, the
antiderivative of v is p.
Now, I want to use the fact that my
velocity at time t is 3 minus 10t.
That means in our specific case that my
position, being the antiderivative of the
velocity and velocity is 3 minus 10t,
right?
So, my position is the antiderivative of
my velocity.
And we can solve that antidifferentiation
problem.
Well, here we go.
Alright?
This is the antiderivative of a
difference, which is the difference of
antiderivatives.
Now, I'd like to antidifferentiate 3,
would be antiderivative of a constant such
that constant times t.
And what's an antiderivative of 10 times
t?
Well, that's 10 times an antiderivative
for t.
So, I got 3t minus 10 times.
What's an antiderivative for t?
By the power rule, an antiderivative for t
is t squared over 2.
And if I differentiate this, I get back t.
I'm going to include a plus C here.
So I can rewrite this as 3t minus 5t
squared plus some constant C.
And now, that plus C has a perfectly
reasonable physical interpretation.
If I know my velocity, I know my position
as long as I know my initial position, as
long as I know where I started, right?
And that's what that plus C is encoding.
If I knew for instance that my position at
times 0 were say 4 units, then I could
figure out what C would have to be right.
That would tell me my position at some
time t, be 3t minus 5t squared plus 4.
This is an example where knowing my
initial position, my position at time 0,
and knowing my velocity for all time, is
enough to figure out my position.
In very fancy terms, this is an example of
solving a differential equation.
I started with the derivative information,
and by antidifferentiating, I recovered
the original function, up to a constant.