A Boeing 747 can travel up to 254m/s. The pilot must travel from Vancouver to Calgary, which is located 3.0 x 10^2 km [N] and 7.0 x 10^2 km [E]. There is a strong wind blowing at 50.0 m/s [SE]
A) determine the direction that the plane must travel to reach its destination in the shortest amount of time?
B) how long will it take for the plane to reach its destination (ignoring take offs/ landings)
To solve this problem, we need to consider both the speed and direction of the plane and the speed and direction of the wind.
Let's start with calculating the effective ground speed of the plane, which takes into account both the speed of the plane and the wind. To calculate the ground speed, we can use vector addition.
1) Finding the ground speed:
Since the plane is flying at a speed of 254 m/s, we need to calculate the resultant vector of the plane's speed and the wind's speed.
The wind speed is given as 50.0 m/s [SE]. We can break it down into its components using trigonometry. SE is 45 degrees between South and East.
Wind speed component in the east direction = wind speed * cos(45 degrees)
Wind speed component in the south direction = wind speed * sin(45 degrees)
Wind speed component in the east direction = 50.0 m/s * cos(45 degrees) = 50.0 m/s * 0.707 = 35.4 m/s
Wind speed component in the south direction = 50.0 m/s * sin(45 degrees) = 50.0 m/s * 0.707 = 35.4 m/s
Now, we can calculate the effective ground speed of the plane by adding the plane's speed and the wind's speed components in each direction.
Effective ground speed (in the east direction) = plane speed (east component) + wind speed (east component)
Effective ground speed (in the south direction) = plane speed (south component) + wind speed (south component)
Effective ground speed (in the east direction) = 254 m/s + 35.4 m/s = 289.4 m/s
Effective ground speed (in the south direction) = 0 (since the wind is not blowing in this direction)
So, the effective ground speed of the plane is 289.4 m/s [E].
A) Determine the direction the plane must travel to reach its destination in the shortest amount of time:
To find the direction the plane must travel in order to reach its destination in the shortest amount of time, we need to determine the angle between the plane's effective ground speed vector and the vector connecting Vancouver to Calgary.
The displacement from Vancouver to Calgary is given as 3.0 x 10^2 km [N] and 7.0 x 10^2 km [E]. We can represent this displacement as a vector.
Now, using trigonometry, we can calculate the angle between the displacement vector and the effective ground speed vector.
Angle = arctan(displacement (north component) / displacement (east component))
Angle = arctan(3 x 10^2 km / 7 x 10^2 km) = arctan(3/7) = 21.8 degrees
So, the angle between the displacement vector and the effective ground speed vector is 21.8 degrees.
The direction the plane must travel is 289.4 m/s [E] at an angle of 21.8 degrees north of east.
B) How long will it take for the plane to reach its destination (ignoring take-offs/landings):
To find the time it will take for the plane to reach its destination, we can use the formula:
Time = Distance / Speed
The total distance from Vancouver to Calgary is given as 3.0 x 10^2 km. However, we need to convert it into meters since the plane speed is in meters per second.
Distance = 3.0 x 10^2 km * 1000 m/km = 3.0 x 10^5 m
Using the effective ground speed of the plane (289.4 m/s):
Time = Distance / Speed = 3.0 x 10^5 m / 289.4 m/s ≈ 1037 seconds
So, it will take approximately 1037 seconds for the plane to reach its destination, ignoring take-offs and landings.