A boat heads due North for 3.00 km to reach a red buoy. It then changes direction and heads in a direction of 30.0 degrees South of East for a distance of 4.66 km to reach a blue buoy. The boat changes direction again and heads to a green buoy which is 1.00 km due East of the boat’s starting position. What must be the boat’s displacement to go from the blue buoy to the green buoy? Express that displacement as the distance traveled in a direction specified as an angle East or West of South.

Question

A boat heads due North for 3.00 km to reach a red buoy. It then changes direction and heads in a direction of 30.0 degrees South of East for a distance of 4.66 km to reach a blue buoy. The boat changes direction again and heads to a green buoy which is 1.00 km due East of the boat’s starting position. What must be the boat’s displacement to go from the blue buoy to the green buoy? Express that displacement as the distance traveled in a direction specified as an angle East or West of South.

Answer 1

just consider the movement at each stage.

If the starting point is O, then the movements (x,y) are
O to R = (0,3)
R to B = (4.04,-2.33)
So, O to B = (4.04,0.67)
so, B to G at (1,0) involves a movement of (-3.04,-0.67)

Now just find that distance and direction.
remember that the angle z will obey tan(z) = ∆y/∆x

Answer 2

To find the boat's displacement from the blue buoy to the green buoy, we can break down the boat's motion into its North-South and East-West components.

1. North-South Component:
The boat traveled 3.00 km due North to reach the red buoy. Then it changes direction and heads 30.0 degrees South of East for 4.66 km. To find the North-South component, we need to determine the change in latitude.

The North-South distance traveled can be found using trigonometry:

North-South distance = distance * sin(angle)

North-South distance = 4.66 km * sin(30.0 degrees)

North-South distance = 2.33 km

Since the boat initially traveled due North for 3.00 km, the net North-South component is:

Net North-South component = 3.00 km - 2.33 km

Net North-South component = 0.67 km South

2. East-West Component:
The boat changes direction again and heads to a green buoy, which is 1.00 km due East of the boat's starting position. Therefore, the East-West component is simply 1.00 km East.

Overall Displacement:
To find the overall displacement, we combine the North-South and East-West components using the Pythagorean theorem.

Displacement = √(North-South component)^2 + (East-West component)^2

Displacement = √(0.67 km)^2 + (1.00 km)^2

Displacement = √(0.4489 km^2 + 1.00 km^2)

Displacement = √(1.4489 km^2)

Displacement = 1.203 km

Direction:
To determine the direction, we can use trigonometry to find the angle between the displacement vector and the South direction.

Angle = arctan(North-South component / East-West component)

Angle = arctan(0.67 km / 1.00 km)

Angle ≈ 34.99 degrees

Since the angle is measured from the South direction, the angle East or West of South would be:

Angle East or West of South = 180 degrees - 34.99 degrees

Angle East or West of South ≈ 145.01 degrees

Therefore, the boat's displacement from the blue buoy to the green buoy is approximately 1.203 km in a direction specified as 145.01 degrees East or West of South.