Solving for Plane Velocity: Dan's Question

[danago]

Ok, I've got a question. Its probably something really stupid that I am overlooking, but ill ask anyway.

Say i have the question:

if a plane is can travel 75 m/s north (75j i assume?) under normal conditions, what velocity will the pilot need to set if he is to travel directly north when there are winds of 21i+8j m/s blowing.

Since i need to find the velocity which will make the plane travel at 75m/s north, 75j is therefore the resultant. If ai+bj is the velocity the pilot needs to set his engines at, i can say:

75j=ai+bj+21i+8j

I then equate the components, to get:

i 0=a+21
j 75=b+8

From that, i can solve for a and say that a=-21, and if i then substitute that into the equation:

$$75^2=(-21)^2+b^2$$

solve for b and that gives me the velocity vector -21i+72j, which is the correct answer. What I am wondering is, when i equated the components, i got two equations. When i used the first equation (0=a+21) and used the value of a to find the final vector, i get the correct answer, but when i used the value from solving b from 75=b+8, i get the wrong answer.

Thanks,
Dan.

 

[HallsofIvy]

Ok, I've got a question. Its probably something really stupid that I am overlooking, but ill ask anyway.

Say i have the question:

if a plane is can travel 75 m/s north (75j i assume?) under normal conditions, what velocity will the pilot need to set if he is to travel directly north when there are winds of 21i+8j m/s blowing.

Since i need to find the velocity which will make the plane travel at 75m/s north, 75j is therefore the resultant. If ai+bj is the velocity the pilot needs to set his engines at, i can say:

75j=ai+bj+21i+8j

I then equate the components, to get:

i 0=a+21
j 75=b+8
From that, i can solve for a and say that a=-21, and if i then substitute that into the equation:

$$75^2=(-21)^2+b^2$$
solve for b and that gives me the velocity vector -21i+72j, which is the correct answer. What I am wondering is, when i equated the components, i got two equations. When i used the first equation (0=a+21) and used the value of a to find the final vector, i get the correct answer, but when i used the value from solving b from 75=b+8, i get the wrong answer.

Thanks,
Dan.

What was the exact wording of the question? You originally set it up so that the airplane's speed relative to the ground, taking into account both velocity relative to the air and air velocity, was 75 m/s due north. If that is the case then 21+ a= 0, 8+ b= 75, so the velocity relative to the air is -21i+ 67j, is correct.
However, if the airplanes speed relative to the air is to be 75 m/s , Then we must have 21+ a= 0 (so that the velocity vector is due north) and a²+ b²= 75² (so that the airspeed is due north). Those are different problems. Is it the planes speed relative to the air or relative to the ground that is to be 75 m/s?

 

[danago]

im a bit confused, but here's exacly word for word what the question asks:

a helicopter can fly at 75m/s in still air. The pilot wishes to fly from airport A to a second airport B, 300km due north of A. If i is a unit vector due east, and j a unit vector due north, find the velocity vector that the pilot should set and the time the journey will take if there is a wind of 21i+8j blowing?

 

[HallsofIvy]

I would interpret this as: the helicopter has a maximum airspeed of 75 m/s. Flying as fast as he can (i.e. at 75 m/s relative to the air) what velocity vector should the pilot take to go due north. There is no requirement that the actual "speed made good" (i.e. relative to the ground) be 75 m/s. In that case, in order to go due n, the helicopter must be angled so that the net "east-west" (i.e. j) component is 0. That's why, with velocity vector ai+ bj, you have b+ 8= 0 (NOT 21+ a= 0 as we both incorrectly said before). The requirement that the airspeed be 75 m/s gives a²+ b²= 75².

 

[danago]

but with b+8=0, it gives a value of -8 for b. And according to the answers page, the answer is -21i+72j.

 

[HallsofIvy]

Sorry, I got my "east and west" confused with my "north and south"!
If his vector velocity, relative to the air is ai+ bj then his actual velocity relative to the ground is ai+ bj+ 21i+ 8j. That must have no x (east and west) component so we must have a+ 21= 0 and a= -21. NOW do what you were talking about before: a²+ b²= 75² to solve for b. That will give you -21i+ 72j.

 

[danago]

im a bit lost when you say velocity relative to the air and ground. But today i went to a study session with my teacher, and from what she said, and what you said, i understand the question a lot better now. Thanks a lot for the help. Greatly appreciated.

Dan.

 

[HallsofIvy]

An airplane flies "on the wind". That is, it is supported by the air and necessarily goes wherever the air goes! Imagine a toy car moving on a table while you are carrying the table to the side. We can calculate the velocity of the car "relative to the table" but have to add to that the motion of the table itself "relative to the floor" in order to find the motion of the car "relative to the floor".

 

[danago]

ohhhh i see now. Makes sense :)

Thanks for that.