00:01
So in this problem, we've got a bit of a vector and calculus problem.
00:05
So then this is a really good example of a problem where not putting in all the numbers we get given in the question straight away is going to really help with working out how we have the best way to do this is.
00:17
So the first thing to do is to draw a set some axes up.
00:24
So let's just let this be x and let this be y, which is going to be north.
00:34
East, south and west.
00:41
And we're going to do, we're starting at the joint base lewis mccorn, which i'm going to put here as my origin, so this location is zero, zero.
00:54
And we're flying to an undisclosed location.
00:58
I'm going to locate this at d, undisclosed.
01:04
And this is located, we're told, 59 kilometers south and 171 kilometers east so as a vector we can write this as 171 in the top pocket for going in the east direction and then minus 59 for going in the south direction and i give this the units of kilometers and we're going to fly from there to there we're further told that there's a mountain let's give it the position are and it located 56 kilometers east and 40 kilometers south.
02:02
In fact, let me redraw that with a bit more space.
02:11
So this is obviously not to scale.
02:16
So that was 56 kilometers east and 40 kilometers south.
02:32
Okay, so that's the actual vector and we are traveling assumingly on a straight line to our undisclosed location.
02:42
So let's say at some point we are here on our line, and our position is going to be given.
02:48
I'm going to give it a name little r as a vector, which is also a function of time, and that's where we are.
03:01
And the question is, or we're also told that we're flying at a constant speed v, which is equal to 800 kilometers per hour.
03:15
And the question is, how long after we depart will we be closest to this mountain? and so one of the things we're going to care about is this distance here, which i'm going to call d of t, because clearly it's going to change over time.
03:33
And we want to find the point where this distance is the smallest.
03:36
So let's set up this problem.
03:42
So first thing i'm going to do is i'm going to write r as a vector just as abstract components, capital r x and capital r y to make that easier for us.
03:58
And then for d, i'm actually going to write down the unit vector associated with d because that's going to be very useful for us, which is just the vector d divided by its length.
04:15
And i'm just going to write this down as abstract coordinates as well, dx, dy.
04:27
So what's our r of t? what's our position? well, we're told we're travelling at a constant velocity.
04:33
We know we start at zero, zero.
04:36
So we can actually just go ahead and start writing down what that means.
04:39
So r of t, we know at zero is going to be zero.
04:44
And because we're going to at a constant velocity, we know our distance is going to be proportionate to the time we've been travelling for.
04:50
So it's going to be equal to the speed we're travelling at times how long we've been travelling by, for that speed.
04:56
But what about the direction? well, we're heading directly towards our undisclosed location.
05:03
So this direction is going to be towards d.
05:06
So we can just put our unit vector there because we don't want the magnitude to affect it.
05:11
So this is how we can write down our position vector as a function of time.
05:17
So now with that, we answer the question, what does the distance as a function of time between us and the mountain look like? well, the distance between two points is just given sort of by the pythagorean argument.
05:32
So it's equal to the square root of the sum of the squares of the differences of the vector components.
05:44
So let me write that down for you.
05:47
So actually, i'll expand this as well.
05:48
So this is easier for us to understand.
05:50
So it says v times t d x, v times t, d y.
05:59
So for our first components, this is the difference of the x values.
06:02
So we have vtdx minus r x all squared.
06:13
Adding onto this, we have the y component differences, vt, dy minus ry squared.
06:26
And this is what that function is.
06:29
And what do we want? we want to find t such that d of t is minimized.
06:53
So how do we find the minimum of a function? well, we're going to look at its first derivative, or when its first derivative goes to zero.
07:00
So to make this slightly easy for itself, we're first going to look at this function and see what we've got the square root of something quite complicated.
07:07
So let's just imagine what would happen for a minute, if we just handle the function...