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problem 1 4 0 is less than or equal to T is less than or equal to 6 a particle is moving along the x axis the particles position X of T is not explicitly given the velocity of the particle is given by V of T is equal to all of this business right over here the acceleration of the particle is given by a of T is equal to all of this business over here they actually didn't have to give us that because the acceleration is just the derivative of the velocity but and they also give us they don't tell us the position function but they tell us where we start off X of 0 is equal to 2 fair enough now let's do Part A is the speed of the particle increasing or decreasing at time T equals 5.5 give a reason for your answer it looks like they did something a little sneaky here because they gave us a velocity function and then they asked about a speed and you might say wait aren't those the same thing and I would say you know they aren't quite the same thing velocity is a magnitude and a direction it is a vector quantity speed is just a magnitude it is a scalar quantity to see the difference you could have a velocity and this isn't maybe particular to this problem because they don't give us the units but you could have a velocity of negative 5 meters per second and maybe if we're talking about on the x axis this would mean removing leftward at 5 meters per second on the x axis so the magnitude is 5 meters per second this is the magnitude magnitude and the direction is specified by the negative number and that is the direction your velocity could be negative 5 meters per second but your speed your speed would just be would just be 5 meters per second so your speed is 5 meters per second whether you're going to the left or to the right your velocity you actually care whether you're going to the left or the right so let's just keep that in mind while we try to solve this problem so the best way to figure out whether our rate of change is increasing or decreasing is to look at the acceleration because acceleration is really just the rate of change of velocity and then we can think a little bit about this this this this velocity versus speed question so what is the acceleration at time five point five get the calculator out we can use calculators for this part of the AP exam and I assume they intend us do because this isn't something to eat that's easy to calculate by hand so the acceleration at time 5.5 we just have to say T is 5.5 and evaluate this function so one-half I'll just write 0.5 times e to the T over 4 well T is five point five five point five divided by four and then times cosine times cosine of times cosine of five point five divided by four five point five divided by 4 gives us gives us 0.38 did I do that right we have 0.5 times e to the 5 point 5 divided by 4 times cosine oh sorry I made a mistake that looked a little strange it's not cosine of 5 point 5 divided by 4 it's cosine of e to the cosine of e to the 5 point 5 divided by 4 so let's look at that so that's one parenthesis I close and now that is the second parenthesis that I've closed and I get negative one point three well to show you right there roughly negative one point three six so this is equal to or approximately equal to negative one point three six if I round it and we don't care about so much as the actual value we get what we really care about is its sine so the acceleration the acceleration at time five point five is negative which tells us that the velocity the velocity is the velocity is decreasing now you might be tempted to say we're done but remember they're not asking us is the velocity of the particle increasing or decreasing they're asking us is the speed is the speed of the particle increasing or decreasing and if you're saying hey how did I know that just remember acceleration is just the rate of change of velocity if acceleration is negative that means the rate of change of velocity is negative it is going down but anyway how do we address this speed issue how do we think about it well there's two scenarios if our velocity is positive at time five point five so if we have a positive velocity so let's say our velocity is five meters per second although they don't give us units here so I won't use units so let's say our velocity is five and then it's negative so at some point our velocity is going to be something smaller then that means that the speed would also be decreasing so if we have a positive velocity then the fact that acceleration is negative means that both velocity and speed would be decreasing on the other hand if we had a negative velocity if we have a negative velocity at time T equals five point five then the fact that it's decreasing means that we're getting even more negative and if we're getting even more negative then that means the speed is increasing the magnitude is increasing in the leftward direction so what we really need to do beyond just evaluating the acceleration at time five point five we also have to evaluate the velocity to see if it's going in the left or in the left or the right word direction so let's evaluate the velocity the velocity at five point five and we'll just get our calculator out again velocity at five point five we need this is our velocity function is going to be equal to two times the sine of let me write it this way just because I want to make sure I get my parentheses right two times the sine actually two times when we write this way to 2 sine of e of e to the 5.5 that's our time divided by four so I did that part right over here I'm going to close the two as well and then plus one so this is our velocity so our velocity at time five point five is negative so negative 0.45 negative 0.45 roughly so this is negative velocity is negative velocity is negative so we have the scenario where the velocity is negative which means we're going in the left direction and the fact that the velocity is also decreasing means that over time at at least at this point in time as we go forward in time it'll become even more negative and it'll become even more negative if we wait a little bit longer so that means that the magnitude of the velocity is increasing it's just going in the leftward direction so if the magnitude of the velocity is increasing although it's going in the leftward direction that means that the speed is increasing so the velocity so this is one of those interesting scenarios the velocity is in decreasing but the speed the speed which is what they're asking us in in the question speed is increasing speed is increasing and if you wanted to do it really quick with all of this explanation I gave you say hey look what's acceleration is it positive or negative you would evaluate it say it's negative so you know velocity is decreasing and then you would just say hey what is velocity is it positive or negative you evaluate you say it's negative so you have a negative value that is decreasing so it's becoming more negative so that means this magnitude is increasing or speed is increasing

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