### Answer

A web page designer creates an computer animation in which a period on a computer system screen has a place of $\vecr=<4.0 \mathrmcm+$ $\left(2.5 \mathrmcm / \mathrms^2\right) t^2 > \hat\boldsymbol\imath+(5.0 \mathrmcm / \mathrms) t \hat\boldsymbolJ$ (a) discover the magnitude and also direction the the dot's average velocity in between $t=0$ and $t=2.0 \mathrms .$ (b) find the magnitude and direction the the instantaneous velocity in ~ $t=0, t=1.0 \mathrms,$ and also $t=2.0 \mathrms .$ (c) sketch the dot's trajectory native $t=0$ to $t=2.0 \mathrms,$ and also show the velocitiescalculated in part (b).

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### Video Transcript

So what we have here is a period on a display given through this position vector right here and for a what we need to do is discover the typical velocity. Uh, in between time equates to zero and time equals two seconds. For this reason the mean velocity is going come be given by the final position, minus the initial place over time. So, uh, because that a time, we No, that is native 0 to 2. So what time's simply gonna be to sex? because that the final and also initial, we're gonna have actually toe look worths in. So the last is in ~ time equals zero. I mean, time you calls to just equal to four centimeters add to t squared, i m sorry is two squared. So 44 times 2.5 is 10 in the I had actually direction, plus five centimeters per 2nd times. Time of two is walking to offer us 10 centimetres in the J have actually direction. Currently our place vector at T amounts to zero. That's going to provide us 4 centimeters in the I had direction plus zero in the J had direction. So now you simply subtract them and you gain 10 centimeters in the I had direction much less 10 centimeters the Jay had actually direction almost everywhere too. And also you can divide to from both those and you're left with 5 centimeters in the I had actually direction in five centimeters in the J a happ direction. So now we uncover the velocity see triangle. For this reason this the I had actually direction. It's the J had direction five and five by the size we provided staggering here. Um, for this reason the magnitude or the median velocity it's going come equal 5 squared plus five squared in other places rooted, and also that is walking to give us an average velocity of 7.1 centimetres she second. Now, to find the magnitude, we just use our trick function. We're, ns mean, no magnitude. Uh, angle. We're looking front to single right here. For this reason we're going to choose Tan data. That cools 5/5 train station tangent of one is data equals 40 5 degrees component B. We require the instantaneous velocities at time equates to zero. Time equals one, and also time amounts to to now to discover the instantaneous velocities. Uh, we have to use the velocity vector, and the velocity vector is going to be provided by the time derivative of our position back. So simply taking the derivative. We have actually the velocity vector same to 2 times 2.55 centimetres per 2nd Squared times, time in the eye cap direction add to jay had actually direction. We just have 5 centimetres per second. So now we simply need to find the velocity. Time equals it's her So us so t in the eye direction That's gonna offer us zero plus just 5 centimetres per second in the J had direction, remember? therefore there's zero in the eye direction. So the means, uh, essentially and we have actually this however this vector right here her this heat is walking to be zero. That method this angle right below is just gonna be 90. Therefore Alfa equates to 90 degrees and also our size of the velocity is 25 senate meters per 2nd for velocity at time equates to one. Us have 5 centimeters. Her 2nd in the I had direction plus five centimeters per second in the J cap direction. Now this is similar, Teoh what we have appear It's the exact same numbers, right? yet it's not the exact same problem, but for self that therefore you know that the direction right here is walk to it is in 45 degrees and also our velocity it's going to equal 7.1 centimetres per 2nd now for the critical one. Velocity of time equals two. We're walking to have 10 centimetres per second. To add in the I had actually direction, the course. 5 centimeters. Her second in the J had direction. Now we have actually to set up ours triangle in the eye direction I had direction is 10 and also in the J have actually direction. Us have 5 looking because that this diagonal. Deserve to we just use private Pythagorean theorem? therefore the V two equal 5 squared to add 10 squared all the square rooted and we finish up through a velocity of 11.2 centimetres per second. Currently I uncover that data we deserve to use tangent tangent or we'll call it Alfa tinge in Alfa equates to opposite over adjacent. For this reason we have that Alfa below is same to 26 0.6 degrees. Currently the last thing we should do is plot the trajectory top top X Y airplane from time zero to time equates to two. And also so what we'll need is the place at time equals zero. Time amounts to one and also time equals to assist us. So in ~ the position, vector the time equals zero. We have four centimeters in the I had direction to add zero in the J cap direction. That means we are starting at 123 four here for there are two me equals one. Us have four plus 2.5. For this reason that's walking to it is in 6.5 centimeters. And I have direction plus five centimeters in the JM direction. That's gonna be 12345 6.5. It will certainly go up to about five se right about there currently or cap Well, T equates to two. So us have 4 plus 2.5 time four. So we have 14 centimeters and also I had actually direction. Plus, she does uncover its 10 centimetres in the J cap directions 6.5789 10 11 14 se Right around here, where it ends up, therefore are lost ease. We have that we have. Therefore you know it goes. The starts at a 90 level trajectory turns and also at this suggest is that 45 and then here it's in ~ 26.6. So right here we have actually the velocity walk upwards. That 5 centimetres per second here is 45 level angle at 7.1 centimetres per second and below it angle.

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Look out 23 26.6. We have actually a velocity 11.2 centimetres every second.