1. A small plane can travel at 200 km/h in still air. If a 50.0 km/h wind is coming from the east, determine the ground velocity of the plane if the pilot keeps the plane pointed [N].

2. A pilot is flying from City A to City B which is 300 km [NW]. If the plane will encounter a constant wind of 80 km/h from the north and the schedule insists that he complete his trip in 0.75 h, what air speed and heading should the plane have?

GOT IT NEVERMIND

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To determine the ground velocity of the plane in the given scenarios, we need to consider the effect of the wind on the plane's motion.

1. For the first scenario, where a small plane can travel at 200 km/h in still air and there is a 50.0 km/h wind coming from the east, the ground velocity of the plane can be found using vector addition.

We can decompose the wind velocity into its north-south and east-west components. Since the wind is coming from the east, the east-west component is 50.0 km/h, and the north-south component is 0 km/h.

To find the ground velocity, we need to add the plane's velocity to the wind velocity components. Since the pilot keeps the plane pointed north (directly against the wind), the plane's velocity is solely in the north-south direction. Therefore, the ground velocity of the plane will be in the north-south direction as well.

The ground velocity of the plane is the sum of the plane's velocity and the wind velocity in the same direction, which is north in this case. So, the ground velocity of the plane will be 200 km/h (plane velocity) + 0 km/h (wind velocity in the north-south direction) = 200 km/h. Therefore, the ground velocity of the plane is 200 km/h, directly north.

2. For the second scenario, where a pilot is flying from City A to City B which is 300 km northwest and there is a constant wind of 80 km/h from the north, we need to determine the airspeed and heading of the plane to complete the trip in 0.75 hours (45 minutes).

To find the airspeed and heading, we can use vector addition again. Since the plane needs to travel northwest, its path will be a combination of north and west directions.

Let's assume the airspeed of the plane is V km/h and the heading angle (angle between the plane's velocity and the north direction) is θ degrees.

Using trigonometry, we can find the north component of the plane's velocity as V * cos(θ) and the west component as V * sin(θ).

To find the ground velocity, we need to add the wind velocity components to the plane's velocity components. The wind velocity is 80 km/h from the north, so its north component is 80 km/h and its west component is 0 km/h.

The ground velocity in the north direction is the sum of the plane's north component and the wind's north component: (V * cos(θ)) + 80 km/h.

The ground velocity in the west direction is the sum of the plane's west component and the wind's west component: V * sin(θ) + 0 km/h.

Since the plane needs to travel northwest, the ground velocity in the west direction should be equal to the ground velocity in the north direction.

From this equation:
V * sin(θ) + 0 km/h = (V * cos(θ)) + 80 km/h.

Simplifying the equation, we find:
V * sin(θ) = V * cos(θ) + 80 km/h.

To solve this equation for V and θ, we can consider different values for V and find the corresponding θ that satisfies the equation.