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(II) An object, which is at the origin at time $t=0,$ has $$\overline{\mathbf{v}}_{0}=(-14.0 \hat{\mathbf{i}}-7.0 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s} $$$$ \quad \overline{\mathbf{a}}=(6.0 \hat{\mathbf{i}}+3.0 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}^{2} . \text { Find the position } \overline{\mathbf{r}} $$ where the object comes to rest (momentarily).

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$\vec{r}=(-16.33 i-8.167) m$

Physics 101 Mechanics

Chapter 3

Kinematics in Two or Three Dimensions; Vectors

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Cornell University

Rutgers, The State University of New Jersey

University of Washington

University of Sheffield

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(II) An object, which is a…

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Okay, So our question states that an object which is at the origin at time t equals zero has a velocity V not equal to minus 14. I have minus seven j happiness per second and acceleration vector equal. The 6.0 I have plus 3.0 j hat meters per square second and then I want you to find the position. Vector are where the object comes to rest momentarily. Okay, so I wrote down what we were given here were given that the initial position is at the origin so are of zero here is equal to zero for all components via zero. Our initial velocity is equal to minus 14. I had minus 17 j hat meters per second. I'm sorry. Not minus 17. It's minus seven. So I can me your rights That one there. So it's minus seven j hat meters per second and then a is equal to 6.0. I had plus 3.0 j hat meters per second squared. Okay, So before we begin solving this problem, we noticed that the position vector can be found from Equation 3-13 B. Since the acceleration vector is constant And the time at which the object is going to come to rest is found by setting the velocity vector equals zero. Both components of the velocity vector must be equal zero at the same time for the object to be at rest. Okay, so first thing we're gonna do is we're gonna write the value of the velocity vector. So the here is equal to Xena, plus the acceleration vector. Time's a good time. Okay, well, this is equal to the knot, which is minus 14 0.0. I have minus 7.0, Jack. Okay. And then that's going to be added to the acceleration, which is 6.0. Go ahead. Bus 3.0 jae ha. That's going to be multiplied by the time now. The object is at rest momentarily when both of'em x. So the ex component is that is equal to zero. Envy of why is equal to zero. Okay, but the X component V of X is minus 14. I had plus six. I had times the time. So you have zero is equal to minus 14. This is the X components will write the sacks out to the side. This is the X component. Okay, zero is equal to minus 14 plus six frank times the time. This is the maximum time for the ex. So we'll just mark this is an X. If you solve for that, you get it T and next is equal to, um it's going to be 14 right? Divided by 6.0 seconds. So C equals 7/3 seconds. Okay. And now for the white part. Well, same thing. It's equal to zero at maximum. Okay, so that means zero is equal to the Y point apart, which is minus 7.0 for the J hat, plus 3.0 times the time. But this one is tea maximum and the Why So we'll do t m y for team maximum in the white Sol for T Max Women, the y can you get the ratio of again? 7/3 seconds. So they're the same. So clearly it's at rest momentarily when the time is equal to 7/3. Okay, now that we know, it's at rest momentarily, when the time is equal to 7/3 we can use our equation here to find the position vector are, which is our not plus the knot of tea. Or maybe not times t plus 1/2 eh Times t squared. Well, t is equal to 7/3 seconds here. Okay? And we also now we just need to plug in our values for a V not. And a which are on the previous page here. Be not his minus 14. I had minus seven J hat. Well, we also need a plug in or value for are not, but that's just zero, so we can ignore that. So, minus 14 I minus seven, Jay had times the time, which is 7/3 okay. Plus 1/2. And then the acceleration, which is sea. Here we go. Back to the first page is six. I have plus three j had Okay, when this is multiplied by the time squared Sosa's 7/3 I swear. Okay. And if you want to put parentheses around all of this and denote this as meters, that's a good idea. Ultra better. And because that is the units of the position vector. Okay, so now if you go through and you carry out all the multiplication so you do minus 14 times 7/3 I had plus 1/2 times six times. 7/3 squared I had That's going to give you minus 16 0.3. I had. Okay, Now we want to find J hats. So you have negative seven J hat time. 7/3 plus 1/2 3 J. Hat times 7/3 squared. That's going to give you minus 8.2. Jay had okay in the units on this meters. Well, go ahead. Boxes in. That's our solution.

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