Ships a and b leave port together. for the next two hours, ship a travels at 28 mph in a direction 32° west of north while ship b travels 24° east of north at 35 mph . -- what is the distance between the two ships two hours after they depart? -- what is the speed of ship a as seen by ship b?

Explanation:

We shall write velocities in vector form

Ship A travels in the direction of 32 °west of north with velocity 28 mph

V₁ = - 28 Sin 32 i + 28 Cos 32 j

Ship B travels in the direction of 24 ° east  of north with velocity 35 mph

V₂ = 35 Sin 24 i + 35 Cos 24 j

Their relative velocity

= V₁ -V₂ =  - 28 Sin 32 i + 28 Cos 32 j -  (35 Sin 24 i + 35 Cos 24 j )

-14.83 i - 14.23 i + 23.74 j - 31.97 j

= - 29.06 i - 8.23 j

Distance between them = relative velocity x time

- 29.06 i - 8.23 j x 2

= - 58.12 i - 16.46 j

magnitude²  =( 58.12 ) ² + ( 16.46)² = 60.40²

magnitude = 60.40 km

Speed of ship A as seen by ship B

= Relative velocity of A wrt B

= - 28 Sin 32 i + 28 Cos 32 j -  (35 Sin 24 i + 35 Cos 24 j )

 =  - 29.06 i - 8.23 j

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