questions-1-2-and-3-are-short-free-response-questions-that-require-about-13-minutes-to-answer-and-ar

Question

Questions 1, 2, and 3 are short free-response questions that require about 13 minutes to answer and are worth 8 points. Questions 4 and 5 are long free-response questions that require about 25 minutes each to answer and are worth 13 points each. Show your work for each part in the space provided after that part.

The motion of an object is given by the following velocity-versus-time graph.
(GRAPH IS NOT AVAILABLE TO COPY)
(a) What is the displacement of the object from time $t=0$ s to $t=6 \mathrm{s}$ in the graph above?
(b) At what times is the speed of the object increasing?
(c) Make of a sketch of the object's position-versus-time graph during the time interval of $t=0$ s to $t=6$ s. Assume that the car begins at $x=0 .$

Vishal Gupta · Accepted Answer

I think there is a velocity time curve as shown here in the figure. The time is represented over the X axis and velocity is represented over the y axis here, this is the origin. If the time is divided In two seconds like one, two, three, four, why, and six, the velocity is one m per second. The velocity is being measured in meters per second and two And in negative direction. Also this is -1 meter per second. No. Here we can do the partitions of the time axis which will be used in plotting the velocity in plotting their displacement time curb. Also, So here these are these partitions and The 6th to 2nd, the last second. And corresponding to this with all city time curb, we will plot the displacement time curve, Which is the 3rd part of the same problem. No, first of all, we plot this velocity time car in which the velocity is one m per second. Initially means when the time is zero, after which the velocity increases to two m/s, Up to a time off to seconds Here, this is two m per second. Now the velocity remains constant as two m per second For the further 1 2nd, then It starts reducing and becomes zero At five seconds. And then In the further 1 2nd The object moves in the opposite direction and the velocity becomes -1 m/s. Now we know the total displacement covered is given by the area covered by VT curb with time access. So displacement is given by the area cover between velocity time curb and the time access. And this is for the first part of the problem. So here the total displacement will be given by the area of the figure Up to two seconds, means this is even and this is a trapeze ium. After that area A two Up to three seconds, which is a rectangle, Then triangle up to five seconds, a three and finally when more triangles, A four up to last six seconds. Now for the area of trapeze um this is half into some of parallel sites and parallel sites are this one m per second and two m per second, so this is one plus two, multiplied by the gap between them, which is from 0 to 2 seconds. So this is two Plus A two area of rectangle, which is the height, which is two m per second, multiplied by the gap between them means three minus two seconds, which is one plus a three. The area of a triangle, right angled triangle, which is given by half into base into height, height again is two m per second and base is five minus three means this is too, and for a four as this is in the fourth quarter and so that will be negative. An area of triangle is having to based This is 1 6 -5 is one, And height is this -1. So Here it comes out to be three plus two plus two and minus 0.5 m. So finally, the total displacement made by this, shown by this vehicle is Given us 6.5 m comes out to be 6.5 m, which is the answer for the first part of the problem. Now, in the second part of the problem, we have to find the time at which the speed is increasing. So it is clear from the curb that speed is increasing initially from zero second to two seconds and then from five seconds to six seconds, but in the opposite direction. Now, in the third part of the problem, we have to convert this Whitaker into displacement time Kirk and foot. Which here we will use the same whitaker and corresponding to it, we are going to plot displacement time curve. Here they are. X axis is again divided into times one second, two seconds, three seconds, four seconds, five seconds and six seconds. And the distance over this, y axis is divided in two m which is one m, two m, three m, four m, five m, six m. Or if he compact the distances in order to accommodate all the distances, this is one, two, three for five, six and seven. Now in the first two seconds the distance covered was three m and that was For their distance from 0-2 seconds. The speed the velocity was increasing, so that will be the current one means it will be curved upward as the speed is increasing, the velocity is increasing like this. Now the velocity becomes constant and the distance covered in the next one second is further more two m. So that will be three plus two means hi meter initially this is three m, not the two m. So there is a connection here, so that is three m in two seconds and that is increasing velocity like this Now beyond two seconds Up to 32nd, The velocity was uniform. So the displacement time curve will be a straight line and extends further two m means it will be like this straight line, it should be a straight line now beyond it. The velocity starts decreasing and the distance covered. Yes, Further two m means up to seven m for five seconds, means of to help and the velocity is decreasing means it will be like this. They&#x27;re decreasing curve whose one decreasing no in the next one second in the last one second, 30 stances 0.5 m and the direction is reversed means it will be 6.5 means here. So the displacement time curve is like this. So this is the overall transformation of viticulture into X. T. Curb here. This is X. And they&#x27;re here. This is deep. Thank you.

Kristela Garcia · Answer

in this problem. We&#x27;re asked to calculate the displacement from 0 to 6 because it is asking for displacement. Anytime the graph goes negative, we do need to consider it as a negative movement. So we do need to subtract it. So here I actually see a triangle, a rectangle, another rectangle, a triangle and then a smaller triangle. So our displacement from a velocity time graph can be calculated by getting the area under the graph. And we say under the graph, we really mean everything from the graft to the X axis and the graph up to the X axis. So for this 1st 1 this area is a rectangle, so this will just be two attempts. One, This is a triangle. So that&#x27;s based on tights or base times, height now our bases to hear. But my height is only one here. So and then because it&#x27;s a triangle based on side divided by two and then this is a rectangle, so base times, height bases. One I is, too. And then this is a triangle. So this is base times height divided by two. So a base of to a height of to and divided by two gives us too. And then this bottom portion here we have also another triangle based on side divided by two are basis one Our height is one and actually negative one here because it is going backwards and we do want to divide by two. So this gives us negative 0.5. So adding all those together we get this one. So we get two plus two plus two plus one, plus a negative 0.5. Is this a positive 6.5 meters And then we&#x27;re asked to you calculate a determined when the speed of the object is increasing while the speed of the object is increasing any time it is moving away from zero on a velocity time graph. So during this segment here is moving away. So it&#x27;s speeding up and then this portion here it&#x27;s also moving away from zero I&#x27;m so it was also speeding up. So it&#x27;s speeding up from section 0 to 2 and 5 to 6 seconds and then part see asks us to create a position time graph. Okay, so this is going to be a statue of a position time grab because I&#x27;m not going to include any numbers, but you could go back and be a little bit more careful about your different displacements because you do have them already calculated. So during the first portion, you are speeding up, and I&#x27;m going to start from a position zero here because it doesn&#x27;t tell me where I started. I can really start from anywhere, but let&#x27;s go ahead. Start position zero. So we&#x27;re gonna speed up for two seconds and the rhythm of constant for one second. And they were going to slow down for two seconds, and then we&#x27;re going to speed up in the negative direction for one second. So we&#x27;re speeding up than moving Constant, then slowing down that speeding up again, Yeah.

Mayukh Banik · Answer

So the X axis is dime axis and the UAE Ecstasy&#x27;s displacement access measured in no, its velocity access measured in made it for a second time is measured in seconds. So that&#x27;s God, he says to for six on eight, 10 20 for eight to have for dinner and 20 now from 57 onwards it is at rest of 20. So five seconds right here. So it is addressed at 20 from five seconds now 01 and two. It is CTO at one. It is four and do its age so that this place spent. So this is the velocity versus time graph. So there&#x27;ll be a bunch of points here. There&#x27;s no point in deciding where the points are exactly. So the displacement of the car between the 1st 2 seconds. So that is the area of this section. So the answer for part A will be half times two times eight. It was eight meter best Barbie. Bart sees displacement during the 1st 4 seconds. So that would be the addition off this part as well. So that is for Barbie and that&#x27;s a This party is for part C. C. One and this body see do so the numbers will be eight meter. Thus the C one will be two over here to over here and eight over there. So that&#x27;s eight times two equals. 16 meter does over here it is. Here is three. That&#x27;s half off it. So that is yes. Times three. No, this distance is Do is the base. Thus one is the top days times the Heidi&#x27;s to elf, So a plus six is 24 meter. Thus six plus trees, 18 meter. So all together the displacement in the 1st 4 second ease for the two major the displacement of the car between Enter eight second Testa answer for party disease. 42 meter I&#x27;m is 4th 2nd to us 20 times 8 to 4 or 4 to 8. So that&#x27;s for the two meter. Thus 80 meter. It was who on 22 meter the slow, big win. The answer. For part a would be find the Silver Design Redoute equals zero and people&#x27;s or so that will be at four second up with our cities eight meters per second. So that is the eight minus zero over that dime ease four second meter per second on the new murder. And second and a denominator on this is two meter for a second square, and this is average acceleration for time. T equals zero. Toe P equals four we&#x27;re about If that&#x27;s no big win. The cake was five to dig was seven. Is Zito on? This means that it is moving at constant velocity.

Ze-Han Lee · Answer

right. So in this problem ah were given Ah, the relation of the velocity of the car versus time. All right. Some party I want to describe. What happened to the car? A timetable to zero. So as we can see, that T equals zero. Ah, border. Yeah, at table 20 would see that the velocity begin decreasing. Right. So take a tickle zero. So the car start digs that already. Okay, so cars started extent awaiting a teak with one and part B. How does the cars Average velocity is in. Ah, tickle. What? He was zero to teach with one competitive average velocity between t equal one and two for five. So from t equal. Ah, so part B from T equals zero to tickle one. Because he&#x27;s a receipt that every velocities over here. Right? So it&#x27;s 10 meters per second. Be bars 10 meters with second from 0 to 1 and from 1 to 5. So again we see that it&#x27;s right here, right? It&#x27;s just the middle off the highest position. Lowest position, which is over here. So it&#x27;s also 10 leave our equal two meters per second. So this is from 1 to 5. So in that sense, the average of lost city off this two individuals of the same. And Parsi says that what is the displacement of the car from 0 to 7? So Ah, because we basically have three stages. The first stages from zero want Ray. So the displacement from 01 that say it&#x27;s x 01 equal, uh, half so t is from 01 race. So it&#x27;s one and times Ah, the ah. So basically, we&#x27;re just calculating the country in the area of this triangle, right? So times 20. So this gives it two meters, and then the second stage is from 1 to 5. So when the culture of the area of this triangle okay, x 15 equal half times so five minus one and times 20. All right, so this gives you 40 meters in. Ah, there is from 5 to 7 when the calculated this area. All right, so x 57 a quote because, as you can see, that the velocity is negative in this area, right? So the area here is also negative, so it equal half negative half, and the time is from 75 to 77 line is fine and times 20. Sorry, snow. Twenties 10. Sorry. Here. Times 10. So this gives you Ah, you see, Negative 10 meters. So totally displacement will be equal to the summation of this three. All right, X equal XO one plus x 15 plus x +57 So this equal 40 meters. All right. And, ah, in the next part. Ah, we want to pluck the car&#x27;s acceleration during the interval as a function of time. Ah, hold on. I&#x27;ll use another one. Another blank paper. Upland diagram. So it&#x27;s right here. So, uh Okay, So full acceleration. I&#x27;ll use the red color. So as you can see that too. The acceleration from 01 is 20 meters. So a ah a one equal 20 meters second squared. Right. So And it is counsel int in this interval. So let&#x27;s say this is a This is 20. This is a 20 and from 1 to 5, a 15 this equal. Ah, 20 over five minus want Ray. So this equal five meters per seconds quit. So this is also a constant but a different matter. Yeah, And from flying to seven. We see that accelerations negative right, A 57 equal. Negative. 10 over seven, minus five, Which is negative. Five meters seconds. Quiet, Ray. So it&#x27;s It&#x27;s like this. All right, so this is X generation. His Parsi. Uh, let me double check the Parsi a party. Oh, it&#x27;s part DEA. There&#x27;s a party, and then, ah, we want to draw. We want to make a sketch off the position during the trouble. Ah, as a function of time and assume that the initial position zero. So we used in another color. Ah. So basically see that from 01 it is accelerating. Right? So the ah, the movement is like is like, um, problem. And it is open up. So, like this, right? And from one to five, as we can see, the velocity is also it It&#x27;s ah, it is positive, but it is the thinks that already so the displacement will be like this at the Tiegel five richest the Maxima. Okay. And, uh, from 5 to 7, state velocity becomes negative rain, so the displacement should decrease in this interval. So you mean like this? So this is the diagram off the displacement

Alexander Cheng · Answer

The question here gives us an equation where we have f of tea, which is going to be positioned. Function is equal to two T cube minus 21 t squared plus 60 t between zero and six year. So, first of all, the first part of this question wants us to graph the particular function. So this function between zero and six year is going toe look like the following, where we&#x27;re gonna have a normal cubic graph where if we have our axes like that, it&#x27;s going to go up and then going to go down and then backed up something like this. Um, for part B here, it asked us to find the velocity functions. So if we remember derivatives, we know that we confined weaken transfer rather between position and velocity and acceleration by finding the first derivative on the second derivative, especially for given the first the function of the position. So in this case, it&#x27;s just gonna be the derivative of F so f of tea here of Prime T. Rather is going to be 60 squared minus 20 minus 42 t rather minus 60. So that is going to be the velocity function. Um, for see here. It asked us to determine the velocity acceleration At T equals one. So to find the acceleration we find, um, the second derivative. So that is gonna be 12 t minus 42. So at T is equal toe one here. We know that the velocity here, so f of one is going to be six minus 40 to minus 66 minutes, 40 to minus 60. That is going to be if I plug this into my calculator. Um, this is gonna be negative. 96 on the second derivative, which is going to be our acceleration in this particular case is just gonna be 12 minus 42. So this is gonna be negative. 30 here. Um, parts Dehere asked us to determine the acceleration when the velocity is equal to zero. So if the velocity is going to be equal to zero, so that&#x27;s gonna be zero is equal to 60 squared minus 40 to t minus 60. We have to solve for the values of T. So t is gonna be equal to 8.217 or negative 1.217 So the only feasible, um value here, which is in between our ranges, is going to be 8.217 So that is going to the time at which the velocity is zero. So based on here, we can plug it back into our acceleration equation, which is gonna be 12 times 8.217 minus 42. So if I put this in 12 times 8.217 on we subtract this by 42 is going to give us a acceleration of 56.60 Um, and lastly, here it asked at which intervals is a speed increasing its all of this. We essentially need to graph our velocity equation, as that tells us whether or not, um our actual speed itself is increasing. So if we graph it, we can see that our speed is gonna be increasing from, um, time of 8.217 onwards.

Veronica Pillar · Answer

This is Chapter two problem eat. So in this problem gives us the equation at the top, this X equals 34 plus 10 T minus two t cubed and asked us to find a few things. So first it asks us to graff x the position of the function of time. Um, And to do that, you could stick that in your calculator and see what comes out. We can also do it more directly by plugging in a few different values of teen About zero seconds, one second, two seconds, three seconds And seeing what you get for X ray to those so for t equals zero. Um, the equation of the top becomes X equals 34 plus zero minus zero, so X equals 34. So let&#x27;s America Point appear for t equals one equation becomes X equals 34 plus 10 minus two. So that gives us X equals 42 meters points right up here. Sorry. I wanted that three blue there. Really, for X ingles, too. We get X equals 34 plus 20 minus 16. So that&#x27;s X equals 38 meters. That points gonna be about here, and her X equals three, We have X equals 34 plus 30 minus 54. So that&#x27;s gonna be 64. Minus 54 is 10 meters away down here. And we know this is Ah, nice. Probably no meal. That should be a smooth curves. I was gonna look something like this. You guys are all the little bumps. All right, so that&#x27;s our answer. Departed with clotted axe is a function of tea from ti ful 0 to 3 seconds in the next part. Wants us to find the average velocity of the object during this time range, and average velocity is given. Bye. Call this fee average total displacement. So that&#x27;s the change in position from the start to finish divided by the elapsed times, the change in time from start to finish. So what are these in this case, Um, our displacement we solved for is gonna be our final position, minus our initial position. And so we just got those values. We just said that our final position was 10 meters, so 10 meters. And we said that our initial position at teeth all zeros 34 meters. So minus 34 meters, we&#x27;re gonna have a negative displacement here and then we divide by the elapsed time, that&#x27;s gonna be three seconds minus zero seconds. So just put three seconds cause that simpler. And if you do out the arithmetic, you get minus 24 meters in the top three seconds on the bottom. So our average velocity is negative eight meters per second. It you really should be eat 8.0 to get significant figures. Right? So that&#x27;s our average velocity. And lastly, it asked us to find at what time is the instantaneous velocity zero. Now, remember, that velocity is the slope of your position versus time graph, so you can see that over here. Initially, the slope is positive. So instantaneous velocities a positive and sometime later, the slope is downward. It&#x27;s negative. So the instantaneous velocity is negative. Somewhere in the middle, there must be a zero instantaneous velocity, and it&#x27;s right where the slope is. Zero. So, from our graph, it looks like that somewhere between t equals one and T equals two seconds, and we can find that precisely by remembering that velocity instantaneous velocity is the derivative with respect the time of the position. So instantaneous velocity, it&#x27;s gonna be DX DT. And so now we got actually take that derivative. And since we&#x27;re given X is a function of time, this will just be a straightforward Catholics. Operation derivative of 34. A constant is just zero derivative the 10 t term. Take the derivative. You just get 10 and the minus toot. He cute becomes a minus. Excuse me is not an eight. It&#x27;ll become a minus six. Pull down. The three was appointed by the two minus six t squared. So this here&#x27;s our formula for instantaneous velocity at all times, and we want to solve for the tie in which this equal zero so 10 minus 60 squared equals zero. We can rearrange that. Move the 60 squared over to the other side and then divide both sides by six. We get 10 over six or in other words, 5/3 equals t squared. And then you just take the square root of both sides. We&#x27;re gonna take the positive square root. Of course, because we&#x27;re looking for a positive time in the interval, if easier on three seconds. And let&#x27;s do that square root of this. I swear to that and you will find that one 0.29 seconds. It&#x27;s a square root of this 10 or a six, and that is going to be the time after which your instantaneous velocity is zero.

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Questions 1, 2, and 3 are short free-response que…

Problem 1 Easy Difficulty

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