(III) You are given a vector in the xy plane that has a magnit...

Question

(III) You are given a vector in the xy plane that has a magnitude of 95.0 units and a y component of -60.0 units. (a) What are the two possibilities for its 𝓍 component? (b) Assuming the 𝓍 component is known to be positive, specify the vector which, if you add it to the original one, would give a resultant vector that is 80.0 units long and points entirely in the - 𝓍 direction.

Accepted Answer

Hey, everyone in this problem, we're asked to consider a velocity vector in the xy plane with a magnitude of 12 m per second. We're told that the X component of the vector's velocity is negative 7 m per second. And the first thing we're asked to do is to find the two possible value sports Y component. Hey, we're gonna stop there for just now. There's a second part to this question. We're gonna come back to that in just a minute. If we take a look at our answer choices, we have four of them A through D option A and B have plus or minus 9.7 m per second for this first part. And option C and D have plus or minus 14 m per second for this first part. So let's draw out what's going on here. That's gonna help us figure out what these values are that we're looking for. We're gonna draw out our plane and we know that the X component of our vector is negative 7 m per second and we know it has a magnitude of 12. OK. So what that means is that our vector could look something like this where the Y component is positive and that vector points up into the left or it could point down into the left with that same magnitude. OK. Both of these vectors have a magnitude of 12, an X component of negative seven. Let's just draw a little better. There we go an X component of negative seven. OK. So they both satisfy all of the information we were given. Now, in order to find the Y component, we can think of this as a triangle and this is a right angle triangle. We know that the hypotenuse is 12, we know that the one side length has a magnitude of seven, we wanna find the other side. So for part one, the magnitude of our rector, we're gonna call M squared is going to be equal to the X component squared plus the Y component squared. Substituting in our values 12 squared is going to be equal to negative seven squared plus Y squared. 144 is equal to 49 plus Y squared. Y squared must then be equal to 95. And when we take the square root, remember that we get both the positive and negative root and this is where those two possible answers come from. So we get that Y is going to be plus or minus the square root of 95. If we use our calculator, we can find that this is gonna be approximately equal to plus or minus 9.7468. And so if we round to two significant digits like the answer choice, we're gonna have that the Y component can be plus or minus 9.7. Ok? So we're looking at option A or B here, we can eliminate options C and D. Now let's go to the second part of this question. So in the second part, they're saying to assume the Y component velocity is confirmed as the positive value. OK. We wanna identify the extra velocity vector needed in order to alter the direction vector to be along the positive Y axis with a magnitude of 20 m per second. OK. So let's break this down bit by bit. OK. That's a lot of wording in that question. So let's kind of write out what that means for the man. So we wanna take our original velocity vector which we're gonna call V one and we're gonna add some additional velocity vector which we're gonna call ve OK. This is our extra vector and this is gonna form some vector that we want in the plane. Now, we're told that the vectors direction is going to be along the positive Y axis. What that means is that the I hat component of the vector is going to be zero this vector points to the along the y axis. Let's go back to our diagram and actually draw this out. So it's gonna point along the y axis with a magnitude of 20. OK. So the X component of the vector is zero and because it has a magnitude of 20 it's all directed along the Y axis, the Y component, the J hat component of this vector is going to be 20. OK. So we have that the vector that we want to make is going to be zero I hat plus 20 J. But now what's vector V one vector V one negative 7 m per second, I hat plus 9.7468 m per second J hat. And we're using plus here because we were told that we confirmed we're taking the positive value the positive Y value there vector ve let's say it has a X component VX I have plus Y component, VY J OK? So we're just breaking these up into components and we know that this is gonna be zero I hat plus 20 J hat. Now, when we're adding vectors, we add by components. So we add the I hat components together that gives us the I hat component of the resulting vector. And we do the same for the J OK. So for the IH components, we have negative seven and we have VX so negative seven meters per second plus VX I ha and then we do the same for the J hat components. So we're adding 9.7468 m per second plus VYJ hat and this half the equal zero, I hat plus 20 J hat. Well, if this has to, the vector on the left has to equal the vector on the right, then the I hat components must be equal and the J hat components must be equal. That's the only way for these vectors to be the same. So we must have that negative 7 m per second plus VX is equal to zero. And we must also have that 9.7468 m per second plus Vy is equal to 20. And this gives us two equations and each of them having one unknown value and we can solve. From the first equation, we get that the X component of a rector V must be 7 m per second. And from the second equation, we get that the Y component must be 10.2532 m per second. So this tells us that that final vector V that we were looking for A or VE if we want to label it as that extra vector is going to be seven, I had plus 10.2532 J hat. And all of this is in meters per second. OK. So that's the extra vector we need in order to create the velocity vector we were interested in. Let's head back to our answer choices. We had already narrowed it down to option A or B. And now we can see that the correct answer is going to be option B so we had that the possible Y components were plus or minus 9.7 m per second. And when we choose that positive value and we look for the extra velocity vector needed to alter our direction. The vector is seven I hat plus 10 J hat meters per second. Thanks everyone for watching. I hope this video helps see you in the next one.

Key Concepts

Vector Components

Pythagorean Theorem in Vector Analysis

Resultant Vectors

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