. A boat is travelling on a bearing of 25° east of north at a speed of 5 knots (a

knot is 1.852 km/h). After travelling for 3 hours, the boat heading is changed
to 180° and it travels for a further 2 hours at 5 knots. What is the boat’s
bearing from its original position?

Just draw a diagram and lay out the vectors.

Start at (0,0)
In 3 hours it has gone 3*1.852km at N 25° E.

So, at that point, its position is (5.55 sin 25°, 5.55 cos 25°) = (2.348,5.035)

Now, it travels 1.852*2 km at S, for another displacement of (0,-3.704)

Its final position is thus (2.348,1.331), which is 3.248km at E 29.5° N (or, N 60.5° E) from (0,0)

If the following forces are applied to a body, what additional force is

required to keep the body in balance:
50 N acting 40° east of north
30 N acting east
60 N acting 20° north of west
15 N acting south

To solve this problem, we can break it down into two parts: the boat's initial travel and its subsequent travel after the change in heading.

Part 1: Initial Travel
1. Convert the speed of 5 knots to km/h: 5 knots * 1.852 km/h = 9.26 km/h.
2. Calculate the distance travelled during the first 3 hours: distance = speed * time = 9.26 km/h * 3 hours = 27.78 km.
3. Draw a diagram indicating the boat's initial bearing, which is 25° east of north, using the information from the question.

N
|
25°|.....Boat
|
|
|____________________

Part 2: Subsequent Travel
4. Calculate the distance travelled during the next 2 hours: distance = speed * time = 9.26 km/h * 2 hours = 18.52 km.
5. Draw a diagram indicating the boat's changed heading, which is 180°, and the subsequent distance travelled.

N
|
|........_____ 18.52 km
|........| | |
|........| | |
|........|.....Boat
|........| |
|........| |
|........|___|

6. Calculate the total distance travelled: 27.78 km + 18.52 km = 46.3 km.

Now, let's find the boat's bearing from its original position.

7. Draw a diagram to represent the boat's final position relative to its original position.

N
|
|
|
|
|
|
|__________Boat
46.3 km

8. Use trigonometry to find the angle between the horizontal and the line connecting the boat's original and final positions.
tan(theta) = opposite/adjacent = 46.3 km/27.78 km = 1.6666
θ = tan^(-1)(1.6666) ≈ 58.23°

9. Finally, subtract this angle from 180° to get the boat's bearing from its original position: 180° - 58.23° ≈ 121.77°.

Hence, the boat's bearing from its original position is approximately 121.77°.

To find the boat's bearing from its original position, we can break down the problem into two parts: the boat's initial path and the boat's final path.

1. Boat's Initial Path:
The boat travels on a bearing of 25° east of north for 3 hours at a speed of 5 knots. Since a knot is 1.852 km/h, the boat's speed is 5 x 1.852 = 9.26 km/h.

To calculate the distance covered by the boat on its initial path, we use the formula: distance = speed x time.
Therefore, the distance covered on its initial path is 9.26 km/h x 3 hours = 27.78 km.

Next, we need to find the boat's coordinates after it travels on its initial path. We can break down the boat's speed into two components: the northward component and the eastward component.

The northward component can be calculated using the formula: northward component = speed x sine(bearing).
So, the northward component on the initial path is 9.26 km/h x sine(25°).

The eastward component can be calculated using the formula: eastward component = speed x cosine(bearing).
So, the eastward component on the initial path is 9.26 km/h x cosine(25°).

Now, we have the northward and eastward components on the initial path. We can calculate the boat's coordinates using the Pythagorean theorem:
distance^2 = northward component^2 + eastward component^2.

2. Boat's Final Path:
After changing its heading to 180°, the boat travels for 2 hours at a speed of 5 knots. The speed remains the same, so the boat's speed is still 9.26 km/h.

Using the same calculations as before, we can find the northward and eastward components on the final path. Let's denote the northward component as N' and the eastward component as E'.

Now, we have the coordinates of the boat after it travels on its final path.

To find the boat's bearing from its original position, we need to calculate the angle between the line connecting its initial position to its final position and the north direction. We can use the arctan function to find this angle:

angle = arctan(E' / N').

This angle will give us the bearing from its original position.