JennyWren
13-10-2006 00:45:13
This first thing you need to realize about this problem is that it is a vector sum. In this case, a vector has a velocity and a direction. This can be represented by a vector in x & y (giving a total velocity vector).
For example, if you were to go straight in the y direction, 1 km/h, your vector would be
0
1 in units of km/h (imagine big square brackets around there!)
Similarly, if you went straight in the x direction, let's say at 2 km/h, our vector would be
2
0
There are three vectors involved in this problem, each with a velocity and a direction. The first two are the velocity from the wind (122 km/h in a direction of south-east) and the total velocity from point a to b (875 km in 1.5 h).
First lets make the units of the total velocity into km/h
875km/1.5h = 583.33 km/h
The total velocity vector is the sum of the wind plus the velocity from the pilot flying the plane. We want to find the velocity from the pilot flying the plane, so we need to subtract the wind velocity vector from the total velocity vector. Let's think about the total velocity vector, and represent it like we did above. Consider NORTH to the the positive Y direction, and EAST to be the positive X direction. Then, the total velocity vector is
0
583.33
in units of km/hr
Now let's try to represent the wind velocity vector the same way. We know the wind is going south-east (because it is coming from the north-west). Because this vector is 45 degrees from the compass axes (also our x, y axes), we know that the change in x and y are going to be the same. X is changing in the POSITIVE direction (because it is going to the east) while y is changing in the NEGATIVE direction (because it is going to the south)
change in x = 122 km/h li sin (45 degrees) = 86.27 km/h
change in y = - 122 km/h li cos (45 degrees) = -86.27 km/h
So the wind velocity vector is
86.27
-86.27
To find the direction the pilot must fly, simple subtract the x and y components of the wind vector from the total vector
0 - 86.27
583.33 -86.27
=
-86.27
669.60
To find the total velocity of the pilot's vector, use pythagoras
velocity = sqrt ( (-86.27li-86.27) + (669.60li669.60) )
= about 675 km/h
To find the direction
angle = arctan ( 86.27 / 669.60 ) = 7.34 degrees to the west of due north.
Anyhow, I hope this helps, here's some diagrams...
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For example, if you were to go straight in the y direction, 1 km/h, your vector would be
0
1 in units of km/h (imagine big square brackets around there!)
Similarly, if you went straight in the x direction, let's say at 2 km/h, our vector would be
2
0
There are three vectors involved in this problem, each with a velocity and a direction. The first two are the velocity from the wind (122 km/h in a direction of south-east) and the total velocity from point a to b (875 km in 1.5 h).
First lets make the units of the total velocity into km/h
875km/1.5h = 583.33 km/h
The total velocity vector is the sum of the wind plus the velocity from the pilot flying the plane. We want to find the velocity from the pilot flying the plane, so we need to subtract the wind velocity vector from the total velocity vector. Let's think about the total velocity vector, and represent it like we did above. Consider NORTH to the the positive Y direction, and EAST to be the positive X direction. Then, the total velocity vector is
0
583.33
in units of km/hr
Now let's try to represent the wind velocity vector the same way. We know the wind is going south-east (because it is coming from the north-west). Because this vector is 45 degrees from the compass axes (also our x, y axes), we know that the change in x and y are going to be the same. X is changing in the POSITIVE direction (because it is going to the east) while y is changing in the NEGATIVE direction (because it is going to the south)
change in x = 122 km/h li sin (45 degrees) = 86.27 km/h
change in y = - 122 km/h li cos (45 degrees) = -86.27 km/h
So the wind velocity vector is
86.27
-86.27
To find the direction the pilot must fly, simple subtract the x and y components of the wind vector from the total vector
0 - 86.27
583.33 -86.27
=
-86.27
669.60
To find the total velocity of the pilot's vector, use pythagoras
velocity = sqrt ( (-86.27li-86.27) + (669.60li669.60) )
= about 675 km/h
To find the direction
angle = arctan ( 86.27 / 669.60 ) = 7.34 degrees to the west of due north.
Anyhow, I hope this helps, here's some diagrams...