Good morning. The momentum version of Newton’s Second Law is net force equals the derivative
of momentum with respect to time where both force and momentum are vectors.
Bo, please rearrange that equation and take the integral of both sides.
Sure. ♪ Flipping Physics ♪
Multiply both sides by dt. Take the integral of both sides. The left side is from time
initial to time final. The right side is from momentum initial to momentum final.
That means we get momentum final minus momentum initial on the right side
which equals change in momentum. That is impulse!
You just derived the equation for impulse! We already knew the equation for impulse.
But you just derived it, which is cool. That is correct Billy. Bo just derived
the equation for impulse. Now, the symbol for impulse is a capital J.
Why not capital I? Capital I is for rotational inertia.
Or moment of inertia. They are the same thing.
Yeah. Okay, capital J for impulse.
Okay, please recall that the Impulse Approximation states that, often, the force of impact is so much
larger than all the other forces acting on the object that we can consider the force
of impact to equal the net force. Therefore, according to impulse approximation, we can
replace the net force in our impulse equation with the force of impact during the collision.
And, usually the symbol used for the force of impact during the collision is just capital
F. Alright, there are many things I need for you to remember about impulse. Impulse is a vector.
That means, class, impulse has both … Magnitude and direction.
Correct. Also, I will just reiterate that the symbol for impulse is not capital I, it is capital
J. I’m repeating that because students so often forget that. Typically, that sounds like this:
Mr.p. What does capital J stand for? Or this:
Dude, what’s capital J? Or even this:
Uh. There’s a capital J on the equation sheet. Does J stand for joules?
So, please remember. The symbol for impulse is … Capital J.
Yep. There are 3 equivalent expressions impulse is equal to. Class, please tell me all three.
Impulse equals change in momentum. Impulse equals the integral from time initial to
time final of impact force with respect to time. (And the third?)
Oh, the area “under” a force vs. time function. Right. That’s the area definition of an integral.
Area “under” the function where area above the horizontal axis is positive
and area under the horizontal axis is negative. Correct. And, Bo, what are the units for impulse?
Well Impulse equals the definite integral of force with respect to time. So, newtons times seconds.
And newtons are kilograms times meters per second squared, which means newton seconds
are the same as kilograms times meters per second which are the units for momentum.
Right. But realize, typically we use newton seconds for impulse and kilogram meters per second
for momentum. It helps delineate the difference between impulse and momentum. Bobby, what is the
integral equation for the definition of work? The integral equation for the definition of
work. Well, that’s not the dot product of force and displacement, that’s for a constant force. Uh
It’s … work equals the definite integral of force with respect to position.
Yes, thank you Bobby. Please notice how similar this integral equation for work is to the integral
equation for impulse. Clearly, they are not the same, however, students often confuse the two.
Work integral equation is with respect to position. Impulse integral equation
is with respect to time. Got it mr.p! Alright, let’s take a look at a typical
force as a function of time graph for a collision between this dropped, rubber,
playground ball and this force platform. First let’s watch it at real speed.
And now let’s watch it at 40 times slower than real speed.
And, while we wait for the rubber ball to actually collide with the force platform again,
let’s take a look at the force of impact as a function of time graph. You can see all of
the data points create a function roughly in the shape of a concave down parabola.
And, notice that the total time during the collision is less than 25 thousandths of a second.
25 thousandths of a second is not very long. 25 milliseconds. Woah.
Yeah. Remember that the area “under” this data is
the impulse during the collision. Also, remember that the area under the curve can be approximated
using a rectangle with a vertical side equal to the average force of impact during the collision
and the horizontal side equal to the change in time during the collision. In other words,
impulse also equals average force times change in time. So yeah, there are a lot of equations
for impulse. Your job is to keep track of all of them and know when to use which one.
To review, Impulse during a collision, the symbol for which is a capital J and is a vector,
equals change in momentum which equals the definite integral of force with respect to time
which equals the area “under” the force vs. time function which equals the average impact force
times the change in time during the collision. And that, of course, does NOT equal work
because work equals the definite integral of force with respect to position
not time. Thank you very much for learning with me today, I enjoyed learning with you.