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# Motion problems: when a particle is speeding up

## Video transcript

let's say that we have some particle that's moving along the number line so let me draw a number line right over here so that's our number line right over there now let's say it starts right over here at zero and then as time passes this little point is going to move around maybe it moves to the right slows down speeds up maybe moves to the left slows down speeds up it might do all sorts of things and to describe this motion its position as a function of time we have a function s of T this particles position as a function of time we're given is T to the third power minus six T squared plus nine T and we're going to restrict the domain to positive time so we're going to assume the time is greater than or equal to zero now the question that we want to answer in this video is when is this particle speeding up so when are we speeding up speeding speeding up and I think that bears a little clarification what does it mean to speed up well there's two scenarios if the particle is already moving in the rightward direction so if it's already moving in the right or directions and the way that we would know it's moving in the rightward direction is if its velocity is greater than zero if it's moving in the rightward direction and it's also accelerating in the rightward direction so if its acceleration is also greater than zero then this is a situation where we are speeding up now another scenario where we would be speeding up is if we're moving in the leftward direction in that case our velocity is going to be negative so if our velocity is negative and we want to go faster in the negative direction then our acceleration should also be negative we're going to that would make our velocity getting more and more and more negative with time so then our acceleration needs to also be negative if we still if we still want to be speeding up if you have any other combination here if your velocity is negative but your acceleration is positive that means that you're becoming less your velocity is becoming less negative or you'd be slowing down in the left direction and vice-versa so if your velocity is positive in your acceleration is negative that means you are going to the right but you are slowing down in the right or directions so let's think about these two scenarios and since velocity matters here so much we just have to remind ourselves that the velocity the velocity remember a derivative is just the rate of change rate of change with respect to a variable so if you have your position function the derivative of position with respect to time this is really just how what is the instantaneous rate of change of position with respect to time well what is change of position with respect to time well that is just going to be that's going to be equal to our velocity function that's going to be equal to our velocity function V of T or we could write that s prime of T s prime of T which could be also written this way as DS DT is equal to our velocity as a function of time so let's take the derivative of this our velocity as a function of time is going to be equal to C 3 T squared 3 T squared minus 12t minus 12t plus 9 plus 9 so let's see if we can if we can graph this velocity function to start making sense of it whether when is the velocity positive when is it negative and what's the acceleration doing in those intervals and so to help me graph it we already we can find that that we could say the the the V intercept or the vertical intercept when V of 0 is going to be equal to 9 so that'll help us graph it that's where we intersect the vertical axis but also let's plot let's plot let's figure out where it intersects the where it intersects the T axis so let's set this equal to 0 so 3t squared minus 12t plus 9 is equal to 0 see to simplify this I can divide both sides by 3 and I get T squared minus 4t plus 3 is equal to 0 now this is very factorable this is T let's see what what two numbers when you take a product get 3 and when you add them you get negative 4 well it's going to be t minus 3 times t minus 1 is equal to 0 how can this expression be equal to 0 well if either of these are equal to 0 if either t minus 3 0 or t minus 1 is 0 - equal to zero so T could be equal to three or T could be equal to one if T is three or T is one either of these are equal to zero and or this entire expression up here is going to be equal to zero and since our coefficient on the T squared term is positive we know this is going to be an upward-opening parabola so let's see if we can plot if we can plot velocity as a function of time so that is my velocity axis this right over here is my time axis time axis and let's say this is one time is 1 second or I'm assuming this is in seconds 2 3 4 actually let me spread them apart a little bit more just because 1 & 3 are significant 1 2 & 3 & is they're not going to be I'm going to squash the vertical scale a little bit but this right over here let's say that is 9 a velocity of 9 and so when T equals zero our velocity is 9 when T equals 1 then this that our velocity is going to be zero this is we get that right over here 3 minus 12 plus 9 that's zero and when T is equal to 3 our velocity is zero again our vertex is going to be right in between those when T is equal to two right in between these two zeros and we could figure out what that what that velocity is if we like 3 it's going to be 3 times 4 minus 12 times 2 plus 9 so what is that that's 12 minus 24 plus 9 so that is negative 12 plus 9 so that's going to be equal to negative 3 did I do that 12 yet negative 3 so you're going to be negative 3 might be that's 9 so that's positive so it might be something like this so the graph of our velocity is a function of time it's going to look something like this we only care about positive time it's going to look something like this so let's think remember this is velocity this is our velocity as a function of time now let's think about when is the velocity less than zero and the acceleration is less than zero so let's think about this case right over here when is this the case both of them are going to be less than zero well velocity is less than zero over this entire interval over this entire interval this whole entire magenta interval but the acceleration isn't zero isn't less than zero that entire time remember the acceleration is the rate of change of velocity we can write we can write here that acceleration as a function of time this is equal to the rate of velocity with respect to the rate which velocity changes with respect to time or we could write acceleration is equal to V prime of T which is the same thing as the second derivative of position with respect to time and so if the acceleration you could really think of the slope of the tangent line of the velocity function and so over here the place where this is downward-sloping where this has a negative slope and the curve itself is below the T axis that's only over that's only over this interval right over here between between this zero right over here and and the and the vertex and we get to this point right over here and then R then our slope flattens out so this interval right over here is T is it's going to be greater than one and it is going to be less than two that meets these constraints now let's think about where our velocity is greater than zero and our acceleration is greater than zero well our velocity is greater than zero over here but notice our acceleration the slope here is negative we're downward sloping so that doesn't apply here our velocity is greater than zero and our the slope of the velocity the rate of change of velocity the acceleration is also greater than zero so that's this interval that's this interval right over here or we're speeding up in the rightward direction so that interval is T T is greater than three so when are we speeding up we're speeding up between the first and second seconds and then we're speeding up after after we're speeding up after the third second