An Accelerating Spaceship?

I'm not quite sure of the question.

First, if the ship continues to accelerate at a heading of "north" at 1 m/s^2, and also accelerating east (perpendicular) at 0.1 m/s^2, there will be no change. This is because you have one fixed coordinate. Your net acceleration will be:

a = a.n i + a.e j

a = 1 m/s^2 i + 0.1 m/s^2 j

a = √ (1^2 + .1^2), at a heading of θ = atan 0.1/1 east of north

a = 1.005 m/s at a heading of θ = 5.7° east of north

However, if you are always changing your orientation and not accelerating "north", but straight, you will go in a spiral relative to the space station.

Under this situation, you will always have a tangential acceleration (relative to the direction of travel) of 1 m/s^2 and a radial acceleration of 0.1 m/s^2

This means that your absolute velocity would be

v(t) = a.tan *t

And the direction would be

dθ(t) / dt = a.rad / v(t)

θ = a.rad/a.tan * ln t

Note: θ is the direction of orientation of the ship. Not the direction that it is in relationship to the space station.

to completely change direction, i.e.,

θ = π

10 π = ln t

t = e^10π

t = 4.4 x 10^13 s

I didn't get into relativity. You would have to reexamine it because your ship would be traveling at a speed of 1.5 x10^5 c -- tachyon speeds. Ref: http://en.wikipedia.org/wiki/Tachyon

***************Addendum**********

There is also a third way to look at this problem which is the rotation of the space ship. This means that space ship would not be accelerating in the direction of travel (per the coordinates of the space station), but in the direction the space ship is pointed.

Here,

dθ /dt = a.rad/a.tan

θ = a.rad/a.tan t

θ = 1/10 t

For θ = π

t = 10 π = 31.4159 s

This might be the more likely, but less interesting, answer.

Interesting problem... I'm tracking this one, hopefully somebody can solve it, be that you, me, or somebody else. If you get it, post the solution.

Right now I'm thinking to describe the position of the spaceship in terms of two functions, x(t) and y(t). The "turning" of the ship due to the perpendicular acceleration has some sort of sinusoidal acc/dec on the initial "straight" acc of the ship. The solution will be when the velocity of the ship in the x direction (I'm assuming he starts out headed for +y) is 0. The motion will be periodic so there will be multiple solutions, starting at t=0, then again at some t=x. We'd be looking for the second occurrence, the one after t=0. The total distance might be the square of the sum of the squares of the integrals of velocity, but I'm just making that up and would be happy to be able to write position equations.

**Edit**

I noticed the same thing about the ship never turning. The way the problem is worded, the ship will merely continue to accelerate forward, and accelerate whichever side direction it chooses. However, I assumed the problem meant that the ship's nose will always point in the direction of the vector sum. So the ship is continually rotated as the 0.1 thrust is applied. This doesn't seem quite right, but it's the only way I can think to make the problem work.

I don't think I can adequately answer this without defining

a central force field and initial conditions. i.e. r(ro,θ,φ). Given this,

we can setup the differential equation and find the time required

for an angular displacement of π radians as you suggested;

I envision the ship as having a tangential acceleration

of ±0.1 m/s^2 and radial acceleration of ±1 m/s^2 relative

to g creating a coplanar spiral orbit (in or out) depending on

how we want to set this up.

PS:

First Grade Rocks!, aka the victim:), seems to be on the

right track.