- [Voiceover] There's a miniature rocketship
and it's full of tiny aliens that just got done
investigating a new moon with lunar pools
and all kinds of organic new life forms.
But they're done investigating, so they're gonna blast off
and take their findings home to tell all their friends.
Let's say at some moment during their ascent,
they're moving at four meters per second.
And they're tiny aliens,
their spaceship is only 2.9 kilograms,
but they need to know, are they gonna be able
to get off this moon or not?
So they gotta pay attention to their speed,
but instead of using a speedometer,
they clever aliens, they use a force versus time graph.
So on their dashboard, they've got a force versus time graph
and it tells them what the net force is on them,
so let's say this is the net force, not just any force,
but this is the total force on them from rocket boosters
and the force of gravity and whatever other forces
there might be, they've got advanced force sensors.
I mean come on, they can determine their net force,
let's say, and it gives them this force
as a function of time.
But they want to know what is their velocity gonna be
after nine seconds?
So they check their force versus time readout,
and this is the graph they get.
And now they can determine it.
And here's how they do it.
They say, "All right there's a force, a net force
"of three newtons acting for the first four seconds."
So during this entire first four seconds
there's a constant force of three newtons.
And every alien worth his weight knows
that force, the net force multiplied by the time duration
during which that force is applied,
gives you the net impulse.
So this gives us the net impulse.
If we take this constant three newtons that acts,
multiplied by four seconds during which it acts,
we get that there's an impulse of 12 newton seconds.
And you might be like, "Wait, who cares
"about newton seconds here, I want the velocity.
"I don't want the force and the time,
"I want to know the velocity at nine seconds."
But they teach you at the alien space academy,
that the net impulse is not only equal
to the net force times the time,
it's also equal to the change in momentum of the object
that the force was exerted on.
And this is good.
We know the mass of the object.
We wanna know something about velocity.
So we know momentum is M times V,
this net impulse is gonna help us get there.
But his 12 newton seconds was only
for the first four seconds.
How do we figure it out for the next three seconds?
Look at this.
During the next three seconds, there's not a constant force,
this force is varying, the force is getting smaller.
So how do I do this?
The force isn't a constant value,
so I can't just simply take force times time,
because, I mean, what force do I pick?
So we're gonna use a trick.
We're gonna use a trick, because if you notice
for this first section, for the first four seconds,
we took the force and multiplied
by the time interval, four seconds.
So what we did really is we just took
the height of this rectangle times
the width of this rectangle
and that gives us the area of this rectangle.
So what we really did is we found the area
under the force versus time graph.
That gave us the impulse and that's not a coincidence.
The impulse equals the area under a force versus time graph,
and this is extremely useful to know,
because now in this section, where the force was varying,
we can still use this, we can just find the impulse
by determining the area under that curve.
And by area under the curve, we mean from the line curve,
in general, to the x-axis, which, in this case,
the x-axis is of the time axis.
So let's do this.
We found the impulse for this first section.
That was 12 newton seconds.
Now we can find the impulse for this next section
by just determining the area.
So this is a triangle.
We'll do 1/2 base, the base is one, two, three seconds,
and the height is still three newtons.
So we get a net impulse of 4.5 newton seconds.
So we've got one more section to go,
but this one's a little weird, this one's located,
the area is located below the time axis,
so this is still a triangle,
but since the forces are negative,
this is gonna count as a negative net impulse.
So when the area lies above the time axis,
it counts as a positive impulse,
and when the area lies below the time axis,
it counts as a negative net impulse.
So how much negative net impulse?
We still find the area.
So the area of a triangle, again, is gonna be 1/2,
the base this time is two seconds,
and the height is negative two.
So negative two newtons, which gives us a net impulse
of negative two newton seconds.
And now we can figure out the velocity
of the spaceship at nine seconds.
So, assuming that this force readout started at this moment
right over here at t equals zero seconds
was the moment when it was going four meters per second,
then we could say the total net impulse should equal
the total change in momentum of this spaceship.
And we can find the total net impulse
by just adding up all the individual impulses.
So during the first four seconds,
there was 12 newton seconds of impulse.
During the next three seconds
there was 4.5 newton seconds of impulse.
And during this last portion
there was negative two newton seconds of impulse,
which if you add all of those up,
12 plus 4.5 plus negative two, you're gonna get
positive 14.5 newton seconds of impulse.
That's good news for our alien buddies over here,
they need to get off this moon,
which means they need positive impulse, upward impulse.
They got some positive impulse.
Let's see what their final velocity was.
We know that delta P is the change in momentum,
so this is final momentum minus initial momentum,
which we could write as
mass times v final minus mass times v initial.
Now, if this were an earth rocket, this would be hard,
'cause earth rockets using earth technology,
eject fuel at a huge rate out the backend,
and that loses mass.
That means this mass isn't gonna stay constant.
So earth rockets essentially push fuel down
which causes and equal and opposite force
back on the rocket upward.
But if you're losing mass, this mass doesn't stay constant,
and this whole process is a lot harder,
because m final and m initial aren't gonna be the same.
Maybe this is where the phrase, "It's not rocket science"
comes from, 'cause rocket science is a little harder
when that mass changes.
So let's just say, these clever aliens can eject
only a little bit of fuel, they do so.
You might say, "How?"
Well, there's gotta be a certain amount of momentum, right,
that they eject to give themselves momentum up.
But let's say they can eject only a small mass
at a huge speed, so there's not much fuel
that they're losing.
The fuel that they eject is ejected at enormous speed,
so that they get their momentum upward,
but they lose almost no mass.
And that let's us solve this problem
assuming that the mass is constant.
So if we do that, if we assume the mass is constant,
we get positive 14.5 newton seconds equals,
we can pull the mass out, the mass is a constant,
so we could just write it as
m times v final minus v initial,
since I can pull out a common factor of m,
which means I can write this as 2.9 kilograms
multiplied by the final velocity after nine seconds
and I know it's after nine seconds, 'cause I added up
all the impulse during the nine seconds,
minus the initial velocity was four meters per second.
So if I divide both sides by 2.9 kilograms,
14.5 over 2.9 is 5 and that will be newton seconds
over kilograms which has units of meters per second,
and it's positive.
And that's gonna equal the leftover over here
is gonna be v final minus four meters per second.
And now finally if I add four meters per second
to both sides, I get the v final somewhere up here,
v final of this rocket is gonna be nine meters per second.
So after nine seconds
it ended up going nine meters per second.
That's just a numerical coincidence.
The way you find it, so recapping the way we did this,
we found the area under the curve,
'cause the area under a curve
under our force versus time graph represents
the impulse on the object.
We found it for the entire trip,
noting that underneath the time axis,
when this curve goes under the time axis,
the net impulse is gonna be negative.
We added up all the net impulse,
we set it equal to the change in momentum,
we plugged in our values and we solved
for our velocity after nine seconds.