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

Planar motion example: acceleration vector | Advanced derivatives | AP Calculus BC | Khan Academy


3m read
·Nov 11, 2024

A particle moves in the XY plane so that at any time ( T ) is greater than or equal to zero, its position vector is given. They provide us the X component and the Y component of our position vectors, and they're both functions of time. What is the particle's acceleration vector at time ( T = 3 )?

All right, so our position, let's denote that it's a vector-valued function. It's going to be a function of time; it is a vector. They already told us that the X component of our position is ( -3T^3 + 4T^2 ) and the Y component is ( T^3 + 2 ). So you give me any time greater than or equal to zero, I put it in here, and I can give you the corresponding X and Y components.

This is one form of notation for a vector. Another way of writing this, you might be familiar with engineering notation, it might be written like:

[
\mathbf{R}(T) = -3T^3 \mathbf{i} + 4T^2 \mathbf{j}
]

or sometimes people write this as unit vector notation:

[
-3T^3 \mathbf{u_x} + 4T^2 \mathbf{u_y}
]

This is just denoting the same thing. This is the X component; this is the Y component. This is a component in the horizontal direction; this is a component in the vertical direction, or the Y component.

Now, the key realization is if you have the position vector, well, the velocity vector is just going to be the derivative of that. So, ( \mathbf{V}(T) ) is just going to be equal to ( \mathbf{R}'(T) ), which is going to be equal to... well, you just have to take the corresponding derivatives of each of the components.

So let's do that. If we want to take the derivative of the X component here with respect to time, we're just going to use the power rule a bunch. So it's ( 3 \times -3 ), so it's ( -9T^2 ) and then plus ( 2 \times 4 = 8 ), so plus ( 8T ).

Then, over here for the Y component, the derivative of ( T^3 ) with respect to ( T ) is ( 3T^2 ), and the derivative of 2 is just zero. So actually, I have space to write that: ( 3T^2 ).

All right, and if we want to find the acceleration function, or the vector-valued function that gives us acceleration as a function of time, well, that's just going to be the derivative of the velocity function with respect to time.

So, this is going to be equal to... let me give myself some space. The X component, well, I just take the derivative of the X component again. Let me find a color I haven't used yet; I'll use this green.

So let's see: ( 2 \times 9 = 18T ) raised to the 1st power plus 8. The derivative of ( 8T ) is just 8 if we're taking the derivative with respect to ( T ). And then here in the orange, the derivative of ( 3T^2 ) using the power rule here over and over again gives us ( 2 \times 3 = 6T ).

So, we've just been able to find the acceleration function by taking the derivative of this position vector-valued function twice. Now, I just have to evaluate it at ( T = 3 ).

So, our acceleration at ( T = 3 ) is equal to: in green, it's going to be ( -9 \times 3^2 + 8 ), and then we're going to have ( 6 \times 3 ).

So what does this simplify to? Well, this is going to be equal to... let's see: ( -9 \times 3^2 = -81 ) and ( -81 + 8 = -73 ). Then for the Y component, we have ( 6 \times 3 = 18 ).

Did I do that arithmetic right? So this is ( -81 + 8 ), which would be ( -73 ), and ( 18 ) stays the same.

Yep, there you have it: the acceleration vector at ( T = 3 ) is:

[
(-73, 18)
]

That is its acceleration. That is its acceleration vector at ( T = 3 ).

More Articles

View All
Populations, communities, and ecosystems | Middle school biology | Khan Academy
In biology, it’s useful to have some shared language so we can communicate and describe the world around us in ways that we can all understand together. So here, we’re going to talk about populations, communities, and ecosystems, and as we’ll see, these …
Conditions for a z test about a proportion | AP Statistics | Khan Academy
[Instructor] Jules works on a small team of 40 employees. Each employee receives an annual rating, the best of which is exceeds expectations. Management claimed that 10% of employees earn this rating, but Jules suspected it was actually less common. She o…
The Child Mind Institute on supporting children during Covid-19 | Homeroom with Sal
Hi everyone, welcome to the daily homeroom! Uh, for those of you all who aren’t familiar with what this is or might just be showing up off of Facebook or YouTube, uh, this is Khan Academy’s way of making sure that we all stay connected during school clos…
How I Built 7 Income Streams at 23 That Retired My Parents
Everyone’s talking about building multiple income streams, jumping side hustle to side Hustle, but here’s what nobody’s telling you: In today’s AI driven economy, being average at multiple things is actually the riskiest position you can be in. Instead of…
TAOISM | The Art of Not Trying
Those who stand on tiptoes do not stand firmly. Those who rush ahead don’t get very far. Those who try to outshine others dim their own light. — Lao Tzu How can we improve when we stop trying to improve? Many people waste their efforts trying to better …
Privacy Policy
Last Updated: 2024-11-07T15:51:10Z Thank you for choosing https://yego.me for your web service needs. We are committed to protecting your privacy and ensuring transparency about how we collect, use, and share your information. This Privacy Policy outline…