A plane is steering at S83degreesE at an airspeed of 550 km/h. the wind is from N70degreesE at 85 km/h. Determine the ground velocity of plane to one decimal. Use the cosine and sine law.

the vectors to add are

v = (550cos7°,550sin7°)+(85cos20°,85sin20°)

so, plug in the numbers and get the sum. Then just get |v| and its direction

To determine the ground velocity of the plane, we can use the cosine and sine laws.

Step 1: Resolve the velocities of the plane and wind into their components.

The plane is steering at S83degreesE, which means it has a heading of 180° - 83° = 97° measured clockwise from the north. The airspeed of the plane is 550 km/h.
The wind is coming from N70degreesE, which means it has a heading of 180° + 70° = 250° measured clockwise from the north. The wind speed is 85 km/h.

The components of the plane's velocity can be calculated as follows:
Plane's velocity in the east direction (Ve) = Airspeed × cos(97°)
Plane's velocity in the south direction (Vs) = Airspeed × sin(97°)

The components of the wind's velocity can be calculated as follows:
Wind's velocity in the east direction (We) = Wind speed × cos(250°)
Wind's velocity in the north direction (Wn) = Wind speed × sin(250°)

Step 2: Calculate the sum of the velocities in each direction.

The ground velocity of the plane can be determined by adding the velocities of the plane and the wind in each direction.
Ground velocity in the east direction = Ve + We
Ground velocity in the south direction = Vs - Wn

Step 3: Use the Pythagorean theorem to calculate the magnitude of the ground velocity.

The magnitude of the ground velocity is given by:
magnitude of ground velocity = sqrt((ground velocity in the east direction)^2 + (ground velocity in the south direction)^2)

Let's calculate each component and the final ground velocity step-by-step:

Step 1:
Ve = 550 km/h × cos(97°) ≈ 151.4 km/h
Vs = 550 km/h × sin(97°) ≈ -539.4 km/h (negative sign indicates south direction)
We = 85 km/h × cos(250°) ≈ -71.2 km/h (negative sign indicates west direction)
Wn = 85 km/h × sin(250°) ≈ -30.1 km/h (negative sign indicates south direction)

Step 2:
Ground velocity in east direction = Ve + We = 151.4 km/h - 71.2 km/h ≈ 80.2 km/h
Ground velocity in south direction = Vs - Wn = -539.4 km/h - (-30.1 km/h) ≈ -509.3 km/h (negative sign indicates south direction)

Step 3:
Magnitude of ground velocity = sqrt((ground velocity in east direction)^2 + (ground velocity in south direction)^2)
Magnitude of ground velocity = sqrt((80.2 km/h)^2 + (-509.3 km/h)^2) ≈ 515.7 km/h

So, the ground velocity of the plane is approximately 515.7 km/h (rounded to one decimal place).

To determine the ground velocity of the plane, we need to consider the effect of both the plane's airspeed and the wind speed. We can use the cosine and sine laws to calculate the ground velocity.

1. First, let's analyze the components of motion. The plane is steering at an airspeed of 550 km/h towards S83°E. This means the angle between the heading of the plane and its reference north direction is 83°.

2. The wind is coming from N70°E with a speed of 85 km/h. This means the angle between the direction of the wind and its reference north direction is 70°.

3. To find the ground velocity, we need to find the resultant velocity vector by considering the vector addition of the airplane's velocity and the wind velocity.

4. We can split the velocities into their respective north and east components. Let's convert all angles to compass bearings, where north is 0° and east is 90°.

- The airplane velocity has a heading of S83°E (or 180° + 83° = 263°) and an airspeed of 550 km/h.
- The wind velocity has a heading of N70°E (or 70°) and a speed of 85 km/h.

5. Next, we can find the components of both velocities in the north and east directions. To find the component of a velocity in a specific direction, we use the cosine law:

Component in a direction = Velocity × cosine(angle between the direction and velocity vector)

- For the airplane velocity:
- North component = 550 km/h × cosine(263°)
- East component = 550 km/h × sine(263°)

- For the wind velocity:
- North component = 85 km/h × cosine(70°)
- East component = 85 km/h × sine(70°)

6. Now we need to sum the components of both velocities to get the total north and east components:

- Total north component = Sum of the north components of airplane and wind velocity
- Total east component = Sum of the east components of airplane and wind velocity

7. Finally, we can use the Pythagorean theorem to find the magnitude of the resultant velocity vector:

Ground velocity = sqrt((Total north component)^2 + (Total east component)^2)

By following these steps and applying the sine and cosine laws appropriately, you should be able to determine the ground velocity of the plane to one decimal place.