- Good morning.
Let's demonstrate that impulse is the area
under a force versus time curve.
♫ Flipping Physics ♫
Billy, we have already derived the impulse equation
from the force of impact equation.
Could you please do so again?
- Uh, I'm not actually sure where to start.
- Start with the impact force equation.
- Net force equals change in momentum over change in time.
- Where both momentum and force are vectors.
- Okay. Um, and multiply both sides by change in time
to get change in momentum equals net force
times change in time, which equals impulse.
- Where momentum, force, and impulse are vectors.
- Yeah.
- Thank you.
Now, we're going to do this again,
only this time from a calculus perspective.
So, the force of impact equation using calculus is,
the net force equals the derivative of momentum
with respect to time.
And we can multiply both sides by dt to get
dp equals the net force times dt.
And we can take the integral of both sides of the equation.
So the integral from the initial momentum
to the final momentum, with respect to momentum,
is equal to the integral from time initial to time final
of the net force with respect to time.
And the definite integral with respect to momentum
is equal to the change in momentum.
Therefore, impulse is the integral from the time initial
to the time final of the net force with respect to time.
And the key thing to understand here is that the integral
is the area under the curve.
- (Bobby) Mr. P.?
- Yes, Bobby?
- I have no idea what you just did there.
- That's okay. You don't have to understand the calculus.
What you need to understand is that I just showed
that impulse is the area under a force versus time curve.
That's what an integral is.
It's the area under a curve.
- Okay, I can know that.
- So if we drop a ball on a force platform,
we can measure the force exerted on the ball
as a function of time.
The area shown in yellow is the area between the curve
and the time axis, which is the Impulse.
In this example, it is 0.81 Newton seconds.
In an algebra based physics class like this one,
we often approximate the force using an average force
and in pink, the rectangle created using the average force
and the change in time during the collision
has the same area and is also equal to the impulse
during the collision.
Let me say that again to make sure it is clear.
The area in yellow and the area in pink have the same value,
0.81 Newton seconds.
The area in yellow is the area is the area between the curve
and the time axis, and the area in pink is the area
of the rectangle created by the average force
and the change in time during the collision.
In this case, it is 91.8 Newtons times 0.008 seconds,
which is 0.73 Newton seconds with two significant digits,
which is not quite exactly the value for Impulse
we measured using the area under the curve.
Now, I actually tried many different objects
and even a different force sensor, and in every instance
the impulse calculated using the average force
and the change in time was always smaller than the area
under the curve.
Now, I attribute this to the fact that we were trying
to measure the force 1000 times every second.
And this force platform from PASCO probably isn't quite
up to that task.
Though the two impulses are pretty close,
I'd actually really love it if somebody from PASCO
could explain to me exactly why they're not the same.
And it's Muumuu Day.
- (Billy and Bobby) Muumuu Day.
- Actually, it's Mu'umu'u Day. It's a Hawaiian word.
- Of course it is. Mahalo.
- (sighs) Thank you very much for learning with me today.
I enjoyed learning with you.