Will a slight change in angle significantly affect distance?

[Mr.Whatever]

Hi!

I was wondering about this topic. The following assumptions are made:

- area is flat and about 70 miles wide from all direction
- ideal airplane only changes in height (roll, azimuth, pitch stay constant).

So here's the scenario:

If a laser beam is mounted on an airplane and is pointed in a 45 degree angle towards the ground (with the right wing of the plane from a cockpit perspective as zero degrees and the angles ascend in a clockwise manner), then would a slight change in the airplane's height significantly alter the location of the beam on the ground?

I was just curious about this topic, and on paper, I'm actually using elevation angle instead of height. The elevation angle is defined as the angle formed between: 1) a line from the plane to the ground directly underneath it and 2) a line from the plane to the point on the ground where the beam is pointing.

I want to know how to calculate the change of the beam's position when this elevationg angle varies by +/- 1 degree.

I know that the elevation angle is dependent on the airplane's height, and that if the airplane was close to the ground then a +/-1 degree change would not really amount to a significant change in position of the beam. But if the plane was, say, 1000 ft in the ground, then this would be a different story.

Thanks!

 

[Data]

I'm a little confused about your definition for elevation angle: from what you've described, it seems that the laser is always pointed at $45^\circ$ towards the ground, and so the elevation angle should always be $45^\circ$ (that would be the angle between a normal to the ground and the laser beam, assuming the ground's flat). I'm sure I've misinterpreted your problem.

However, using the interpretation that stated above, it's easy to see what happens if the airplane rises or lowers. In this case all the angles stay the same; there is a right triangle with $45^\circ$ corner angles and vertices at (a) the point on the ground directly below the laser, (b) the point at which the laser strikes the ground, and (c) the laser.

In that case increasing altitude corresponds to increasing the distance from (a) to (c); furthermore, because of the 45 degree angles the height of the plane (ie. the distance (a) to (c)) is the same as the distance between (b) and (a).

By similar triangles, scaling the height by a factor $\Delta$ will scale all other distances in the same way. So if you increase the height of the plane by 1%, then the distance between the point directly below the plane on the ground and the point where the laser strikes the ground will also increase by 1% (ie. between (b) and (a)). Because the angles are $45^\circ$, in fact increasing the height by a $h$ will also increase the distance on the ground by $h$ (ie. you don't need to worry about any scaling; the increases in altitude is exactly the increase in distance along the ground).

 

[Gdarnall]

^^ +1 , yes it will and it will change more significantly as you become farther (higher) from the ground. think about it like compound interest * but only if your angles greater than 45 degrees




* these statements have not been evaluated.