Will the Coastguard's Intervention Prevent a Ship Collision?

[phospho]

In this question, the horizontal unit vectors i and j are directed due east and north respectively.

A coastguard station O monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship A is at the point with position vector (−3i+10j)km relative to O and has velocity (2i+2j)kmh−1. Ship B is at the point with position vector (6i+j)km and has velocity (−i+5j)kmh−1.

a Show that if the two ships maintain these velocities they will collide.

The coastguard radios ship A and orders it to reduce its speed to move with velocity (i+j)kmh−1. Given that A obeys this order and maintains this new constant velocity.

b find an expression for the vector AB→ at time t hours after noon,

c find, to three significant figures, the distance between A and B at 1500 hours,

d find the time at which B will be due north of A.

a) done
b) (9-2t)i + (-9+4t)j done
c) done
d) I'm not sure how to do this but I thought if B will be due north of A, then the i components should be the same no? so I done this i.e 10+t = 16 => t = 6 however this is wrong and in the solutions they say that the i component should be equal to 0, however in the next question:


Two ships P and Q are moving along straight lines with constant velocities. Initially P is at a point O and the position vector of Q relative to O is (12i+6j)km, where i and j are unit vectors directed due east and due north respectively. Ship P is moving with velocity 6ikmh−1 and ship Q is moving with velocity (−3i+6j)kmh−1. At time t hours the position vectors of P and Q relative to O are p km and q km respectively.

a Find p and q in terms of t.

b Calculate the distance of Q from P when t=4.

c Calculate the value of t when Q is due north of P.


Part c) they say that it's when the i components are equal, which is what I done for the first question.

Could anyone explain what is going on?

 

[Simon Bridge]

(d) the vector \vec{AB} will only have a \vec{j} component.

 

[Clipper Victor]

How did you get the answer for part b in that 1st question?
I'm stuck on how to represent AB

 

[Simon Bridge]

Welcome to PF;

How did you get the answer for part b in that 1st question?

Vector addition.

I'm stuck on how to represent AB

Represent the vectors by arrows on a bit of graph paper and work it out.
AB is the vector pointing from point A to point B.
How would you normally do that?

 

[Clipper Victor]

I would use i+I notation. But I can't be sure that

 

[Clipper Victor]

Sorry. Message accidentally sent before finished

 

[Clipper Victor]

I can't be sure that the position of the ships haven't changed

 

[Simon Bridge]

The positon of the ships does change - their relative position changes too.
So you need to represent their positions as a vector that changes in time.

i.e. if ship A traveled at speed 3kmph 45deg from the +x axis starting at position $\vec a_0= 3\hat\jmath$km, then it's position vector will be $\vec a = (3\hat\imath + (3+3t)\hat\jmath)$km.

 

[Clipper Victor]

I'm not sure what hat means

 

[Simon Bridge]

indicates a unit vector:
$\hat\imath = \mathbf{i}$, $\hat\jmath = \mathbf{j}$, $\hat k = \mathbf{k}$
... are the Cartesian unit vectors.

In general: $$\hat v = \frac{\vec v}{v}$$

 

[Clipper Victor]

The book doesn't mention hat. It just says that the coastguard radios ship A to reduce speed to i+j kmh-1

 

[Simon Bridge]

Do you know what the i and j mean?

 

[Clipper Victor]

It's fine. I got an answer. Thank you