An aeroplane flies horizontally at 80m/s in still air. If the aviator wishes to fly due south and the wind is blowing from the south-east at 30m/s,
1. what course must he steer.
2. how long will it take him to arrive at his destination 200km away.
To calculate the course the aviator must steer and the time it will take him to reach his destination, we can break down the motion of the airplane into two components: the horizontal motion due to the airplane's velocity, and the horizontal motion due to the wind velocity.
1. To determine the course the aviator must steer, we can use vector addition. Since the airplane wants to fly due south and the wind is blowing from the southeast, the wind can be broken down into its east and north components. The east component of the wind velocity is given by the wind velocity multiplied by the cosine of the angle between the wind direction and the east direction. Similarly, the north component of the wind velocity is given by the wind velocity multiplied by the sine of the angle.
Let's find the components of the wind velocity:
East component = 30 m/s * cos(45°)
North component = 30 m/s * sin(45°)
The east component will act as a vertical correction, and the remaining horizontal component will give us the course the aviator must steer. Since the airplane's velocity is given as 80 m/s in still air, the aviator must steer in a direction that opposes the eastward wind component while maintaining a speed of 80 m/s. This can be represented by a right triangle, with the northward wind component as the vertical side and the difference between the airplane's velocity and the eastward wind component as the horizontal side.
Using the Pythagorean theorem, we can find the magnitude of the course:
course = sqrt((80 m/s - east component)^2 + north component^2)
2. To calculate the time it will take to travel 200 km, we need to find the effective velocity of the airplane. The effective velocity is the magnitude of the airplane's velocity vector, which is the vector sum of the airplane's velocity and the wind velocity.
Let's find the effective velocity:
Horizontal component of velocity = 80 m/s - east component
Vertical component of velocity = north component
Using the Pythagorean theorem, we can find the magnitude of the effective velocity:
effective velocity = sqrt((horizontal component of velocity)^2 + (vertical component of velocity)^2)
To calculate the time, we use the formula:
time = distance / effective velocity
Let's substitute the given values to find the course and the time.
A = direction angle steered east of south
V south = Vs = 80 cos A - 30 cos 45
V east = Ve = 80 sin A - 30 sin 45 = 0 if going south
so
80 sin A = 30* .707
etc