Good morning. Class, how many uniformly accelerated motion variables are there?
Five! How many UAM equations are there?
Four! If you know how many of the UAM variables?
Three! You can figure out the other …?
Two! This leaves you with one?
Happy physics student! I always like that.
I do too. ♪ Flipping Physics ♪
Yeah. Good. Now, I am going to introduce
alternate versions of all four of these equations. The first equation is v equals
v sub zero plus a times t, or velocity equals velocity sub zero times acceleration times time.
There are many differences between this first UAM equation and our original first UAM equation.
Let’s begin by looking at how we have replaced change in time with just the variable time. Bobby,
what does change in time equal? Change in a variable always equals
final minus initial, so change in time equals time final minus time initial.
Correct. Now, we replace time final with the variable time with no subscript and
we assume the initial time is zero, therefore change in time equals just the time variable.
The letter we use for constant acceleration, a, is the same as before. Instead of velocity initial,
v sub i, we have v sub zero or v “naught”. The “naught” here is the digit zero. It means the
velocity at time zero, hence v “naught”. The word “naught” in math means zero.
And we know the initial velocity is the velocity at the initial time, and that initial time equals
zero. We showed that when we showed change in time equals the time variable. Right mr.p?
Correct Billy. And just like we use the time variable without a subscript instead of time
final, we also use the velocity variable without a subscript instead of velocity
final. The second uniformly accelerated motion equation in the new form is this:
Bo, please read the equation and explain to me the differences between this
and our original form of the equation. Our new second UAM equation is position equals
position naught plus velocity naught times time plus one-half acceleration times time squared.
Well, the letter, a, for constant acceleration is again the same as before.
We have the same differences as before; we have replaced velocity initial with velocity naught
and we have replaced change in time with time. Now we have replaced position initial with position
naught and replaced position final with position. But how is that the same as having change in
position on the left side of the equation? Oh, subtract position naught from both
sides of the equation and you have position minus position naught on the left-hand side,
that’s the same thing as position final minus position initial, which is change in position. So,
it’s the same thing. Nice.
Great. Thank you, Bobby and Bo. The third new UAM equation is velocity squared equals
velocity naught squared times time plus two times acceleration times the quantity
position minus position naught. And the fourth UAM equation is position minus
position naught equals one half times the quantity velocity plus velocity naught all times time.
Now, this fourth UAM equation is not on the AP equation sheet for some reason,
however, you are welcome to use it. Okay, Billy, please take the derivative of the
new second UAM equation with respect to time. Sure. The equation is position equals position
naught plus velocity naught times time plus one-half times acceleration times
time squared. And, we are taking the derivative of that equation with respect to time. Well,
let’s start with the derivative of position naught with respect to time.
How do we do that? How do we take the derivative of the initial position with respect to time?
Remember, the initial position is a constant number. The initial position
does not change with respect to time. It’s just the initial position. It’s just a number.
Oh, so the derivative of the initial position or position naught with respect to time equals zero.
Because the derivative of a constant number equals zero, we showed that last time. That makes sense.
Okay. We are done with the first term, position naught. Now let’s take the derivative with respect
to time of velocity naught times time. … That works out to just be velocity
naught. Why?
It’s just like the derivative of 5t with respect to time equals 5. Velocity naught is just a
constant number, like 5. Oh, right. Thanks.
And now the derivative with respect to time of one-half acceleration times time squared. Well,
we know we multiply the term by 2 and raise time to the power 1. The one half is a constant number,
so that just stays at one half. But, what about acceleration, what do we do with that?
Ah, acceleration is constant, just like v naught. So, you just leave it as acceleration.
Right. That is what uniformly accelerated motion means; the acceleration is constant.
So, the derivative of this position equation with respect to time equals velocity naught plus
acceleration times time. Very nice y’all.
What? Uh? Are we missing something? Tell me when you see it.
That’s the first UAM equation. What?
Oh, the derivative of position as a function of time is velocity.
So, we just derived the equation velocity equals velocity naught plus acceleration times time.
Oh. Actually, that equation
tells us that the instantaneous velocity of an object equals the velocity of the object at time
zero plus the acceleration of the object times the time at which the instantaneous velocity occurs.
This is an equation for instantaneous velocity. Correct everyone. Taking the derivative
of the second uniformly accelerated motion equation with respect to time
gives us the first UAM equation. In order to be able to take derivatives like this,
we have these new expressions for the uniformly accelerated motion equations.
Also realize that, using calculus, we begin to see relationships form between equations which,
using only algebra, were more difficult to see before. When we get to integrals,
we will be able to derive all of the UAM equations starting with the definition of
acceleration. Thank you very much for learning with me today, I enjoyed learning with you.