An air ambulance is travelling from Barrie to Toronto. Toronto is located 90 km [S5°E] of Barrie. If the wind is blowing from the South with a velocity of 42 km/h, and the plane’s air speed is 400 km/h, what direction must the pilot fly to make it to Toronto?
fly Theta deg East of South
south speed = 400 cos Theta - 42
east speed = 400 sin Theta
tan Theta = 400 sin Theta / [ 400 cos Theta - 42 ] = tan 5
solve for Theta
To determine the direction the pilot must fly to make it to Toronto, we need to consider the effect of the wind on the plane's course.
Let's break down the given information:
- The plane's airspeed is 400 km/h. This is the speed at which the plane travels relative to the air.
- The wind is blowing from the South with a velocity of 42 km/h. This means the wind is blowing toward the North.
To calculate the ground speed (the speed of the plane over the ground), we need to take into account the wind's effect on the plane's speed and direction.
First, we calculate the component of the wind speed that opposes the plane's motion. Since the wind is blowing from the South and the plane is flying towards the South (S5°E), the component of the wind that opposes the plane's motion is along the plane's course and can be calculated using trigonometry.
The component of the wind opposing the plane's motion = Wind Speed * cosine (angle between the wind direction and the plane's course)
Since the angle between the wind direction (South) and the plane's course (S5°E) is 90° - 5° = 85°, we have:
Component of the wind opposing the plane's motion = 42 km/h *cos(85°) ≈ 5.62 km/h
Now, we can calculate the ground speed of the plane:
Ground Speed = Air Speed - Component of the wind opposing the plane's motion
Ground Speed = 400 km/h - 5.62 km/h
Ground Speed ≈ 394.38 km/h
To determine the direction the pilot must fly, we need to find the angle between the ground speed vector and the desired direction (S5°E). Let's call this angle x.
Using trigonometry, we can find angle x:
cos(x) = Component of the ground speed along the desired direction (S5°E) / Ground Speed
The component of the ground speed along the desired direction can be calculated as follows:
Component of the ground speed along the desired direction = Ground Speed * cosine (angle between the ground speed vector and the desired direction)
Since the angle between the ground speed vector and the desired direction (S5°E) is 5°, we have:
Component of the ground speed along the desired direction = 394.38 km/h * cos(5°) ≈ 393.28 km/h
Now, we can calculate angle x:
cos(x) = 393.28 km/h / 394.38 km/h
x = arccos(393.28 km/h / 394.38 km/h) ≈ 1.802°
Therefore, the pilot must fly a direction of approximately S5°E with a slight correction towards the East (right) by about 1.802° to make it to Toronto.