Good morning. Billy, what is the equation for momentum?
♪ Flipping Physics ♪
Momentum, or lowercase p, equals mass times velocity,
where both momentum and velocity are vectors. Billy, thank you for pointing out that the symbol
for momentum is a lowercase p. And we know not to confuse lowercase p for momentum with uppercase P
which is for power, or lowercase rho, which looks somewhat like a lowercase p,
the symbol used in physics for volumetric density. Bobby, what are the units for momentum?
The units for momentum are well momentum equals mass times velocity, so
kilograms times meters per second. Correct Bobby. And Bo, what is the special name
we have for kilograms times meters per second? Uh …
That is correct Bo, there is no special name for kilogram meters per second. They are just called
kilogram meters per second. Right.
Bo, what is Newton’s Second Law in terms of acceleration?
Newton’s Second Law is net force equals mass times acceleration where
both force and acceleration are vectors. Yep. And we can now introduce a form of
Newton’s Second Law which is closer to the original law, one that involves momentum.
The net force acting on an object, or a system of objects, equals the derivative
of momentum with respect to time, where both force and momentum are vectors. Substituting in
(Hold up mr.p) Yes, Bo?
Um. What do you mean “closer to Newton’s Original Second Law”?
What have we been working with this whole time? Oh, sure. Sorry. Newton’s original 3 laws
were published in 1687 in his book titled “The Mathematical Principles of Natural Philosophy”.
Actually, it was 3 books, and he published them 3 times, 1687, 1713, and 1726, each
time correcting and improving on his theories. Also, it was originally published in Latin. So,
the English translation of Newton’s Second Law from 1687 reads “The alteration of motion is
ever proportional to the motive force impressed; and is made in the direction of the right line in
which that force is impressed.” What? Yeah. Uh?
Right. All that is to say that we do not use his original law in his own words,
we use a modernized version which uses newer terminology and equations.
Right. Yeah. Uh? Anyway, getting back to the matter at hand.
Newton’s Second Law in terms of momentum states that the net force acting on an object or a system
of objects equals the derivative of momentum with respect to time where both force and momentum
are vectors. Billy, please substitute in the equation for momentum and take the derivative.
Okay. Um. Momentum equals mass times velocity. And to take the derivative of that
I’m not sure how we do that. You need to use the product rule.
Right, the derivative of x times y with respect to time equals the derivative of
x with respect to time times y plus x times the derivative of y with respect to time.
Got it. Thanks. So The derivative of mass times velocity with respect to time equals
the derivative of mass with respect to time times velocity plus mass times the derivative
of velocity with respect to time. And the derivative of velocity with
respect to time equals acceleration. That means, the version of Newton’s
Second Law we have been using until now assumes the mass of the object or system
does not change as a function of time. Oh right. It assumes the derivative
of mass with respect to time equals zero. Very nice y’all. Yes, the form of Newton’s
Second Law in terms of acceleration assumes the mass of the object or system does not
change. Which means, using the new version of Newton’s Second Law in terms of momentum,
the mass of the object or system of objects can change. Think, rockets!
Neeto! Cool! Yea. But first, …
when the net force acting on a system equals zero, the derivative of momentum with respect to
time equals zero, therefore the momentum of the system is not changing, therefore the momentum
of the system is conserved. This is called conservation of momentum. The sum of the initial
momenta equals the sum of the final momenta. We have derived conservation of linear momentum.
Got it. Linear momentum of a system is conserved when all the forces are internal to the system
and the net force acting on the system equals zero. Like a rocket. Right mr.p?
Exactly. I feel like we should get to see a rocket.
Yeah. Would you like to see a rocket?
Yes. Already.
Alright. This is a water rocket. The bottle is about one third filled with water. The half inch
pvc pipes and bottle are filled with compressed air. When I pull on the string, the water rocket
will be released, the water will go down, and the rocket will go up. Here we go. Clearly at
real speed that happens pretty quickly, so let’s slow it down to 40 times slower than real speed.
And let’s watch that one more time so we can pause it right here,
to understand the basics of how this water rocket accelerates upward. Please recall my
demonstration of throwing a medicine ball while standing on a skateboard
and my demonstration of throwing the same medicine ball out of a canoe. In those demonstrations
the medicine ball moves to the left with negative momentum, and the skateboard and I, or the canoe
and I, move to the right with positive momentum. This is because momentum is conserved when I throw
the ball, because the force I apply on the ball is equal and opposite to the force the ball applies
on me. In other words, all the forces are internal to each system and the net force on the system
equals zero. The same thing is happening here in the water rocket. Air pressure inside the rocket
is pushing individual molecules of water out the bottom of the rocket, all forces are internal,
momentum is conserved, the water molecules have downward momentum
and the rocket has upward momentum. Each water molecule is just like a microscopic medicine ball
being pushed downward by the rocket causing the rocket to move upward. That is how a rocket works.
And, as the water leaves the rocket, the mass of the rocket decreases.
(mr.p?) Yes Billy?
But the net force acting on the rocket and water system does not equal zero because there is a
force of gravity acting downward on it, so the net force acting on the rocket and water
equals the downward force of gravity. Oh right. Valid point.
We are considering the force of gravity to be small enough relative to the internal forces
acting within the rocket and water such that the force of gravity is negligible.
And you can see that is true because the rocket accelerates upward with quite a large magnitude.
Thanks, that makes sense. So, there you have it. We have the very basics of
how rockets work. Thank you very much for learning with me today, I enjoyed learning with you.