1. Ship A is located 2 km north and 1 km east of Ship B....

1. Ship A is located 2 km north and 1 km east of Ship B. Ship A has a velocity of 3 km/h toward the north, and Ship B has a velocity of 5 km/h in a direction 53° north of east. Recall that sin(53°) = 0.8 and cos(53°) = 0.6. (a) What is the velocity of Ship A relative to Ship B in unit-vector notation with i? toward the east and j? toward the north? (b) Write an expression (in terms of i? and j?) for the position of Ship A relative to Ship B as a function of time t, where t = 0 when the ships are at the positions described above. (c) At what time is the separation between the ships the smallest? (d) The ships must come within 0.5 km of one another in order for the sailors on the ships to communicate using semaphore [signal flags]. Do the ships ever get close enough to do so?

Transcript

00:01 In this problem, you have two ships a and b.

00:04 In the end, our goal is to get the velocity of a relative to b.

00:08 We do have information about how the ships are moving relative to the earth, the shore.

00:17 We also are given the initial separation between the two, one kilometer east.

00:25 A is one kilometer to the east, two kilometers to the north of b.

00:31 Remember, you can think of.

00:33 Of another set of axes where b is the origin.

00:37 So as b moves, you still have this vector and that is relative to b.

00:42 It's to the origin of another coordinate system, just like r -a -e is to the coordinate system of the one fixed to the earth.

00:51 So that's what we mean by relative to.

00:55 Okay, now we are told that ship a, relative to the earth, is moving due north, three kilometers per hour.

01:05 So that's 3km slash h .j head.

01:08 We are told that the ship b is moving to speed 5 kilometers per hour at an angle 53 degrees north of east.

01:15 East is the baseline and it's north of that line.

01:19 So we have to get v -b -e vector.

01:24 So that would be 5 kilometers per hour, cosine 53 degrees.

01:31 I -hat plus 5 kilometers per hour.

01:37 Sign 53 degrees j hat they give you to one significant digits cosine and sign so we'll use those make our life a little easier numerically three kilometers per hour i hat plus four kilometers per hour j hat so we now know both a e and b e and now we can begin the parts for like i said the first thing it wants is what is the velocity of a relative to b.

02:14 Well, how do you do this? well, the idea behind this is the following.

02:20 Think about what goes on.

02:23 Let's leave alone.

02:23 We got a ship here, but let's talk about a ship and a person moving on that ship.

02:27 How do we find what that person is doing relative to the shore? we add as vectors the motion of the person on the ship and what the ship is doing to the shore.

02:37 It's a vector operation, two -dimensional.

02:39 Well, we add as vectors the motion.

02:40 In this case, the two -dimensional vector operation, but it's a vector operation, a vector sum.

02:45 So, in our case, v -a, relative to the earth, is v -a -b, that's what we're looking for, plus v -b -e.

02:59 That's the idea of, so in my case, a is the person, b is the ship, and that would give you the person relative to the shore.

03:11 That would be how my example meant.

03:13 Up, but a vector sum.

03:16 Notice something about this.

03:18 Notice, the b's side by side.

03:23 So you can think of kind of crossing them out, a .e.

03:25 A .e.

03:25 That's how you check yourself.

03:27 You could even do this, to be honest with you.

03:30 You could actually write a, b on this side, and you would have e, b on the other side.

03:38 Well, the earth, relative to b, is just a negative of b relative to the earth.

03:43 So it's the same thing.

03:45 So we can just do some vector operations here.

03:48 A -b -b -a -e minus v -b -e.

03:54 Like i said, if you wanted to use that same, let me show you what i meant, what i was talking about just a minute ago.

04:11 If you wanted to construct this directly, v -a -b, and so you would have here a -e -e -v -e -v -e -factor plus v -a -b, eb to have them side by side.

04:32 But this is minus v, b.

04:37 So if the b is relative to the earth is moving to the right, say, b will say that the earth is moving to the left.

04:47 And that's where the minus sign comes in.

04:49 So you can see, either vector operation or going back to the basics, what we mean by relative motion, and you get the same expression.

04:57 Oh, this is a...

05:02 Now, we have all those vectors.

05:06 So we can just put it together.

05:08 A .b.

05:10 3 kilometers per hour j hat minus.

05:15 And we got these.

05:16 Minus three kilometers per hour i hat minus four kilometers per hour j hat.

05:27 And we can combine here minus three kilometers per hour i hat minus.

05:36 We got three minus four.

05:38 So minus 1 kilometer per hour j hat.

05:43 So that is the velocity of a relative to b.

05:48 Again, that's basically how that's relative to the origin, how that rab vector is changing, which is drawn from the origin of b.

06:01 You know, you think of an axis moving with b, set of axes moving with b, and b is at the origin of that set of axes.

06:08 Okay, that was part.

06:10 A, that's part a, part b.

06:15 Now it wants in terms of the position vector for all time, that's a function of time.

06:22 Now we can be, here it is in vector form, this is what we mean, and if we're looking at this from a standpoint of components, r -a -b -x, i'll just call it x -d -x -a -b, that's going to be from x, a, b, 0 to x -a -b, that's for at any time, is equal to from zero to t, v -a -b -x -d -t...

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