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Current time:0:00Total duration:4:35

AP.CALC:

FUN‑8 (EU)

, FUN‑8.B (LO)

, FUN‑8.B.1 (EK)

a particle moves in the XY plane so that at any time T is greater than or equal to zero its position vector is and they give us the X component and the y component of our position vectors and they're both functions of time what is the particles acceleration vector at time T equals three 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 and they already told us that the X component of our position is negative 3 T to the third power plus 4 T squared and the y component is T to the third power plus 2 and 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 and 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 or sometimes people write this as unit vector notation negative 3 to the third plus 4t squared times the unit vector in the horizontal direction plus T to the third plus 2 times the unit vector in the vertical direction this is just a noting 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 of 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 so V of T is just going to be equal to R prime of 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 so if we want to take the derivative of the X component here with respect to time we're gonna use the power rule a bunch so it's 3 times negative 3 so it's negative 9t squared and then plus 2 times 4 is 8 so plus 8 T to the first so plus 8 T and then and then over here for the Y component so the derivative of T to the third with respect to T is 3 T squared 3 T's and the derivative of two is just a zero so actually have space to write the three T squared even bigger 3t squared all right and if we want to find the acceleration function as a or the 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 this is going to be equal to let me give myself some space and so the x component well I just take the derivative of the X component again and let me find a color I haven't used yet I'll use this green so let's see 2 times negative 9 negative 18 times T to the first power plus 8 derivative of a T is just 8 if we're taking the derivative with respect to T and then and then here in the orange derivative 3t squared so it's 2 power rule here over and over again 2 times 3 is 6 t to the first power just 60 so this is we've just been able to by taking the derivative of a vector value of this position vector valued function twice I'm able to find the acceleration function and now I just have to evaluate it at T equals 3 so our acceleration at T is equal to 3 is equal to so in green it's going to be negative 18 times 3 plus 8 comma comma and then we're gonna have 6 times 3 6 times 3 and so what does this simplify to well this is going to be equal to let's see negative 18 times 3 is negative 54 negative 54 plus 8 is negative 46 negative 46 and 6 times 3 is 18 did I do that arithmetic right so this is negative 54 Plus 8 so negative 54 plus 4 would be negative 50 plus another 4 would be negative 46 yep there you have it negative 46 comma comma 18 that is its acceleration that is his acceleration vector at T is equal to three