An aeroplane can fly at 250 km/h in still air.How long will it take it to reach a plane 500 km to south-east if a wind is blowing at 80 km/h from west?
Draw the diagram, head to tail (wind east, then vector down until it intercepts SE vector, and is length 250.
You have two sides (80,250) and an angle opposite the 250 side.
law of sines:
sin45/250=sinTheta/80 where Theta is the angle the still air vector intercepts the SE movement vector.
Solve Theta.
Now, the final angle, Theta1, the angle between the wind and still air vector
Theta1=180-45-Theta
Now you have the final angle,
ground speed/sinTheta1=250/Sin45
solve for ground speed.
time=500km/groundspeed
To determine how long it will take for the airplane to reach the destination, we need to find the effective speed of the airplane when considering the wind. Here's how we can calculate it step by step:
1. Analyze the wind's effect:
- The wind is blowing at 80 km/h from the west.
- The plane needs to travel 500 km to the southeast.
2. Decompose the wind's velocity into horizontal and vertical components:
- The horizontal component is 80 km/h towards the east (opposite to the plane's direction).
- The vertical component is 0 km/h since the wind is not blowing directly up or down.
3. Calculate the resultant velocity:
- We can use vector addition to find the resultant velocity. The speed of the airplane in still air is 250 km/h, and the wind's horizontal component is -80 km/h (opposite direction of the plane's movement).
- The resultant velocity can be found by adding the airplane's velocity to the wind's velocity:
Resultant velocity = Airplane velocity + Wind velocity
V = 250 km/h + (-80 km/h)
V = 170 km/h (south-east direction)
4. Determine the time taken to cover the distance:
- Now that we know the effective speed of the airplane (170 km/h), we can calculate the time it will take to cover the given distance of 500 km.
- Time = Distance / Speed
Time = 500 km / 170 km/h
Time ≈ 2.94 hours (rounded to two decimal places)
Therefore, it will take approximately 2.94 hours for the airplane to reach the destination.