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
See the Ancient Whale Skull Recovered From a Virginia Swamp | National Geographic
When I first went to the site in the bottom of the river, you see these whale bones and shark teeth just poking out. The river’s raging; it’s like holding on to a car going 65 miles an hour down the highway. Everything east of the Route 95 on the east sid…
Mariana Van Zeller visits Disney Parks for Earth Month | ourHOME | National Geographic
I’m Mariana Vanel, National Geographic’s investigative journalist. I’ve traveled to the deepest corners of our world documenting stories with a global impact. This Earth Month, I’m headed to Walt Disney World Resort to spotlight their sustainability work …
Earth's First Selfie | Generation X
With you watching on a dark December night, the final Apollo mission blasts off. As the astronauts leave Earth behind, they do something remarkable: they take a family photo. As the astronauts were leaving Earth, within just a few hours, they were able to…
Taxing Unrealized Values Can Destroy Billionaires
Most people don’t realize that this can actually make Warren Buffett and Jeff Bezos go broke and send their stocks crashing. The reason is because 48 trillion dollars of stock value equals zero dollars in real money, and the IRS only takes real money. Bi…
Nature's 3D Printer: MIND BLOWING Cocoon in Rainforest - Smarter Every Day 94
Hey, it’s me, Destin. Welcome back to Smarter Every Day! So, we just got off this boat, and we’re gonna walk for about an hour in the jungle to find a moth pupa. Okay, Phil just found it. So, what are we looking at here? This here is the pupa of a moth c…
Why is this number everywhere?
Let me show you something unbelievable. Name a random number between 1 and 100. 61. Okay, that’s pretty random. [Emily] Just name a random number from 1 to 100, random. 43. 43, thank you so much. 56. 7. I want the most random number between 1 and …