yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Motion problems: finding the maximum acceleration | AP Calculus AB | Khan Academy


3m read
·Nov 11, 2024

A particle moves along the x-axis so that at any time T greater than or equal to zero, its velocity is given by ( V(T) = T^3 + 6T^2 + 2T ).

At what value of T does the particle obtain its maximum acceleration? So we want to figure out when it obtains its maximum acceleration.

Let’s just review what they gave us. They gave us velocity as a function of time. So let’s just remind ourselves: if we have, let’s say, our position is a function of time, so let’s say ( X(T) ) is position as a function of time, then if we were to take the derivative of that, ( X'(T) ), well, that’s going to be the rate of change of position with respect to time, or the velocity as a function of time.

If we were to take the derivative of our velocity, then that’s going to be the rate of change of velocity with respect to time—well, that’s going to be acceleration as a function of time. So they give us velocity. From velocity, we can figure out acceleration.

Let me just rewrite that. So we know that ( V(T) = T^3 + 6T^2 + 2T ). From that, we can figure out the acceleration as a function of time, which is just going to be the derivative with respect to T of the velocity.

So just use the power rule a bunch. That’s going to be this is a third power right there: ( 3T^2 + 12T + 2 ). So that’s our acceleration as a function of time. We want to figure out when we obtain our maximum acceleration.

Just inspecting this acceleration function here, we see it's quadratic; it has a second-degree polynomial. We have a negative coefficient out in front of the highest degree term, in front of the quadratic second-degree term, so it is going to be a downward opening parabola.

Let me draw in the same color. So it is going to have that general shape, and it will indeed take on a maximum value. But how do we figure out that maximum value? Well, that maximum value is going to happen when the acceleration value, when the slope of its tangent line is equal to zero.

We could also verify that it is concave downwards at that point using the second derivative test by showing that the second derivative is negative there. So let’s do that; let’s look at the first and second derivatives of our acceleration function.

I’ll switch colors; that one’s actually a little bit hard to see. The first derivative, the rate of change of acceleration, is going to be equal to: so this is ( -6T + 12 ). Now let’s think about when this thing equals zero. Well, if we subtract 12 from both sides, we get ( -6T = -12 ).

Divide both sides by -6; you get ( T = 2 ). So a couple of things: you could just say, “All right, look, I know that this is a downward opening parabola right over here. I have a negative coefficient on my second-degree term. I know that the slope of the tangent line here is zero at ( T = 2 ), so that’s going to be my maximum point.”

Or you could go a little bit further; you can take the second derivative. Let’s do that just for kicks. So we could take the second derivative of our acceleration function. This is going to be equal to 6, right? The derivative of ( -6T ) is 6, and the derivative of a constant is just zero.

So this thing, the second derivative, is always negative. So we are always concave downward. And so by the second derivative test at ( T = 2 ), well, at ( T = 2 ), our second derivative of our acceleration function is going to be negative.

And so we know that this is our maximum value, or max, at ( T = 2 ). So at what value of T does the particle obtain its maximum acceleration? At ( T = 2 ).

More Articles

View All
All Shower Thoughts I Had This Year
have you ever paused to think about how one of the most famous sentences of all time doesn’t make grammatical sense? Well, because we all apparently heard it wrong and continue to say it wrong. According to the man himself, Neil Armstrong, what he did say…
Impact of mutations on translation into amino acids | High school biology | Khan Academy
So let’s start looking at a short sequence of DNA and the letters. I’m going to use these as the shorthands for the various nucleotide bases that make up a sequence of DNA. So let’s say that I have some thymine, thymine, cytosine, guanine, cytosine, thym…
Help support Khan Academy
Hi everyone, Sal Khan here from Khan Academy, and I just wanted to remind you that we are a not-for-profit, and we can only exist through donations from folks like yourself. Our goal is for everyone to reach their potential. Potential is everywhere; unfo…
Colonial Weaponry | Saints & Strangers
[Music] Radio weapons, push off, push off design. Mr. Bradford, fire! This is your standard, uh, standard matchlock musket. It was the earliest firing, uh, musket that there was. This over here is a match cord; both sides were normally kept lit in case …
'Property is theft' stolen concept fallacy
Property is theft. This is a phrase that unpacks as all property is theft, and it’s something that I’ve seen mentioned a few times on YouTube lately. A comment from one of my subscribers, I think in my previous video, prompted me to address this specifica…
Introduction to contractions | The Apostrophe | Punctuation | Khan Academy
Hello grammarians! Hello David! Hello Paige! So today we’re going to talk about contractions, which are another use for our friend the apostrophe. So David, what is a contraction? So something that apostrophes are really good at doing is showing when le…