Question

A ship leaves port at noon at a bearing of 143°. The ship’s average rate of speed over this time is 12 miles per hour.

Describe the location of the ship at 4:30 p.m. by writing a vector in component form. Then explain what these components mean in terms of the scenario.

Answer 1

The location of the ship with respect to the port at 4:30 pm is
`\vec r = -43.126\,i + 32.498\,j\,[mi]`.

Explanation:

The ship is travelling at constant velocity, which means that magnitude and direction of velocity vector remains unchanged in time. Then, position vector with respect to port as a function of time and in component form, that is, rectangular form, is:

`\vec r = x \,i + y \,j`

`\vec r = (r \cdot \cos \theta) \,i + (r\cdot \sin \theta)\,j`

`\vec r = (\dot r \cdot t\cdot \cos \theta) \,i + (\dot r \cdot t \cdot \sin \theta)\,j`

Where
`t` is the time interval with respect to noon, which is measured in hours.

Given that
`\dot r = 12\,(mi)/(h)`,
`\theta = 143^(\circ)` and
`t = 4.5\,h`, the location of the ship at 4:30 pm is:

`\vec r = \left[\left(12\,(mi)/(h) \right)\cdot (4.5\,h)\cdot \cos 143^(\circ) \right]\, i + \left[\left(12\,(mi)/(h) \right)\cdot (4.5\,h)\cdot \sin 143^(\circ) \right]\,j`

`\vec r = -43.126\,i + 32.498\,j\,[mi]`

Answer 2

x and y components are (32.498, -43.1263)

They mean that the ship is 32.498 miles east and 43.1263 miles south

Explanation:

We will first of all need to find the time interval we are considering for the travel.

This is 4:30 pm - 12 noon = 4 hours 40 minutes. or 4.5 hours.

Next, we will find the distance travelled by multiplying its speed by this time.

This is 12 X 4.5 = 54 miles.

Hence the vector is 54 miles on a bearing of 143 degrees.

we will resolve this vector into its x and y components since our answer is required in component form.

The angle we will consider is (143-90) = 53 degrees

x component = (54 cos 53) = 32.498

y component = -(54 sin 53) = -43.1263

x and y components are (32.498, -43.1263)

They mean that the ship is 32.498 miles east and 43.1263 miles south