A spaceship maneuvering near Planet Zeta is located at r = ( 600î ─ 400ĵ + 200k ) X 10³ km, relative to the planet, and traveling at v = 9500î m/s. It turns on its thruster engine and accelerates with a = ( 40î ─ 20k ) m/s² for 35 min. What is the spaceship's position when the engine shuts off? Give your answer as a position vector measured in km.

Video transcript

Hey, everyone in this problem at T equals zero. A submarine is moving at V equals negative eight J hat meters per second. It is located at R knot equals 5000 I hat plus 125 J hat plus 750 K hat in meters with respect to a reference island at T equals zero seconds. The submarine accelerates at a rate of A is equal to 1.25 I hat minus 2. J hat. OK. And meters per second squared for two minutes, we're asked to calculate the submarine's position vector after two minutes of motion. And so we're looking for the position vector. We're given four answer choices. Option A the position is 1.4 I hat minus 1.88 J hat plus 0.75 K hat times 10 to the four m. Option B R is equal to 1.4 I hat plus 1.88 J hat minus 0.75 K hat times 10 to the four m. Option C R is equal to 2.65 I hat minus 1.45 J hat plus K hat Times 10 to the exponent two m. And option D R is equal to 2.65, I hat plus 1.45 J hat minus 75 K hat times 10 to the exponent two m. Now we're looking for a position vector, we're given information about velocity um about acceleration about time. So we want to think about our kinematic equations, but we have three directions to consider OK. We have an I hat component AJ hat component and a K hat component. So we have three directions. So let's write out what we know for the X direction. OK. Or the I hat component. How do we know? Oh the initial speed and it's gonna be VNA X, the initial velocity Is going to be zero m/s. The initial velocity we're given is V is equal to negative eight J hat meters per second. OK? So there's AJ hat component but there's no I hat component. So we also have our Y direction, OK? The J hat direction and R Z direction or the direction OK. While we're looking at V, not, let's write out the components for each. So V not in the Y direction, this is AJ hat component we have is negative eight m per second. And again, we only have AJ hat component. We don't have an I hat component or a Z component. And so V knot Z is going to be zero m per second just like the X direction. OK. So we've done with V what about V F the final velocity? Well, we aren't given any information about the final velocity in any of the directions. So we don't have information on that. What about the change in position? OK. For I hat. So the change in position, delta X, well, this is our final exposition minus our initial exposition. We're looking for our final position. So what we want to find is that final exposition and the final Y position and the final Z position. Now we're given our initial position as a vector and in the component, We have 5000. And so delta X A change in position or the displacement is going to be X F minus that initial position of m. Now we move on to our time And the time we're given is two minutes In order to convert two minutes into seconds, we're gonna multiply by 60 and there are 60 seconds in every minute. And so two minutes is equal to 120 seconds. And the last piece of information we want to look at is the acceleration in the X direction. Now, if we look at our acceleration vector, OK, we wanna look at the X component or the I hat component and the acceleration is 1.25 I hat K minus some J hat component. So our I hat component is 1.25 and the unit is meters per second squared. OK. So we've done the I hack direction or I hat component. Now we're gonna move to the Y direction. OK. We've already done the velocities, the displacement or the change in position, delta Y it's gonna be the final position minus the initial position. Again, we're looking for the final position. So we're gonna need the Y component of that position. The initial physician vector we were given has AJ hat component of a 125. And so our change In position in the Y Direction is gonna be YF - m. And we have the same time of 120 seconds and the acceleration in the Y direction. If we look at the acceleration, we were given, we have 2.5 J hat. And so the Y component, the J hat component is 2.5 m per second squared. OK. So we've got all of our variables written down for the I hat component. The J hat component, let's do the khat component. The change in position again, Z F minus is that not Z F minus our initial position, which is 750 in the K hat direction. So we have Z F minus 750 m. Our time is 120 seconds And the acceleration vector does not have a Khat component. And so the acceleration is just 0m per second squared in that direction. All right. So we have all of our variables written out, we wanna find the position vector. So we're looking for X F Y F N Z F, we're not given any information about the final velocity. So we want to choose the kinematic equation that doesn't include final velocity, but includes all those other variables that we have as well as the displacement vector that we're looking for. For the I hat component, we get the delta X is equal to VA X multiplied by T plus one half multiplied by A X multiplied by T square. Substituting in the information we have, we get X F -5000 m is equal to V is zero V not X. And so this first term goes to zero, We got 1/2 multiplied by 1.25 m per second squared, multiplied by 120 seconds squared. This gives us X F - m Is equal to 9000 m. OK? And solving for X F, we get X F is equaled to 14, meters. OK. So we've found the final exposition. That's one component of what we were looking for. Let's do the same for the Y direction or that J hat component, we're gonna use the exact same equation. So in this case, we have delta Y is equal to V not Y multiplied by T plus one half A Y multiplied by T squared. Subsequuting inner values YF -125 m Is equal to -8 meters per second, Multiplied by 120 seconds plus one half Multiplied by 2.5. Me, let me, I'm going to erase the second term here. I'm gonna write it underneath just so that we don't have to squish it in and we don't run out of room so we get plus one half multiplied by 2. m per second squared, multiplied by 120 seconds squared. A simplifying YF - m Is equal to negative 960 m - m. And we can move the 125 m to the right hand side by adding, and we get the final Y position is negative 18, 835 m. OK? So we've got our final X position, our final Y position, we have one left to do and that's the Z position or the K hat component. Now, this one is a little simpler. We have an acceleration of zero and when we have an acceleration of zero, the equation OK. Governing that motion is going to be simplified. So we can use that same equation. We get delta Z is equal to V not Z multiplied by T that's one half multiplied by A Z multiplied by T squared. OK. The acceleration is zero. So the right hand side, the second term goes to zero. But note that the initial speed or that the initial velocity is also zero. So this entire right hand side goes to zero, we get that delta Z Is equal to zero. Delta Z is Z F -750 m, which tells us that the final position of Z is equal to 750 m. And notice that this is the exact same as the initial position for that Z direction or that K hat component. And that makes sense, we have no initial speed and it's not accelerating. So we shouldn't the position in that component in that direction won't change. So we've got our three components. Now let's write this position vector as a whole. So our position vector R It's going to be equal to 14,000. I have -18835 J. Hat Plus 750 KA. And this is all in meters. Now, we want to write this in scientific notation. So we can rank this as 1. Times 10 to the exponent or I hat -1.8, 8 Times 10 to the exponent for J. Hat. And now we've had, we have 10 to the exponent 4, 10 to the exponent four in our I hat and J hat components, let's write the K A T component with the exact same exponent. Um so that we can factor it out and compare it to our answer choices. So we can write this as plus 0.07, five Times 10 to the exponent for Khat. This is all in meters. And one final step, we can factor out that 10 to the exponent four. We get 1.4 I hat minus 1.88 J hat plus 0.75 K hat Times 10 to the exponent four m. And that is that final position we were looking for. After two minutes, we compare this with their answer choices. We found that the correct position vector is R is equal to 1.4 I hat minus 1.88 J hat plus 0.75 K hat times to the four m which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.