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Vectors, EXT1 V1 SM-Bank 31

A drone is set to fly west at 38 km/h.

A cross wind diverts its path so that it travels with a speed of 45 km/h in the direction shown below. 
 

Calculate the speed, to one decimal place, and the bearing of the cross wind, to the nearest degree.  (3 marks)

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`text{41.9 km/h on a bearing of 022°.}`

Show Worked Solution
 
`underset~u + underset~v` `= (-45 sin 30 \ , \ 45 cos 30)`
  `= (-22.5 \ , \ {45 sqrt3}/{2})`

 
`underset~u = (-38 , 0) \ , \ underset~v = (x , y)`

`((-38),(0)) + ((x), (y))` `= ((-22.5),({45 sqrt3}/{2}))`
`((x), (y))` `= ((15.5),({45 sqrt3}/{2}))`

 

`| underset~v|` `= sqrt{15.5^2 + {45^2 xx 3}/{4}}`  
  `= 41.94…`  
  `= 41.9 \ text{km/h}`  

 

`tan theta` `= {{45 sqrt3}/{2}}/{15.5} = 2.514`
`theta` `= 68.3^@`

 

`:.\ text(The crosswind has a speed of 41.9 km/h on a bearing of 022°.)`

Vectors, EXT1 V1 SM-Bank 30

A man attempts to swim north across a river at a speed of 4.5 km/h.

If the river's current is moving at 11 km/h due east, find the bearing  (nearest degree) and speed (to 1 decimal place) that the man will actually be swimming.  (3 marks)

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`text{11.9 km/h on a bearing of} \ 068^@`

Show Worked Solution

`underset~u = (0, 4.5)`

`underset~v = (11, 0)`

`underset~u  + underset~v = ((0),(4.5)) + ((11),(0)) = ((11),(4.5))`
  

`tan theta`  `= 11/4.5`
`theta` `= tan^{-1} (11/4.5)`
`theta` `= 67.75`
`theta` `~~ 068^@`
   
`| underset~u + underset~v |` `= sqrt{11^2 + 4.5^2}`
  `= 11.88 ..`
  `= 11.9 \ text{km/h}\ \ text{(1 d.p.)}`

 
`:. \ text{Swimmer’s speed is 11.9 km/h on a bearing of} \ 068^@`

Vectors, EXT1 V1 2021 HSC 14a

A plane needs to travel to a destination that is on a bearing of 063°. The engine is set to fly at a constant 175 km/h. However, there is a wind from the south with a constant speed of 42 km/h.

On what constant bearing, to the nearest degree, should the direction of the plane be set in order to reach the destination?  (3 marks)

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`75^@`

Show Worked Solution

`underset~u = (175 sintheta, 175 costheta)`

♦♦ Mean mark 31%.

`underset~v = (0, 42)`

`underset~u + underset~v = (175 sintheta, 175 costheta + 42)`

`:. tan 63° = (175 sintheta)/(175costheta + 42)`

`(175 costheta + 42)tan63°` `= 175 sintheta`
`175costheta tan63° + 42tan63°` `= 175 sintheta`
`175sintheta – 175costheta tan63°` `= 42 tan63°`
`sintheta – costheta\ (sin63°)/(cos63°)` `= 6/25 (sin63°)/(cos63°)`
`sintheta cos63° – costheta sin63°` `= 6/25\ sin63°`
`sin(theta – 63°)` `= 6/25\ sin63°`
`theta – 63°` `= sin^(-1)(6/25\ sin63°)`
`theta – 63°` `= 12°`
`:. theta` `~~ 75°`