# Inertial Coordinate Systems

##### New member
A train moves due West at a constant speed v. It passes a carousel two
mile due South of the carousel at 1:00 p.m. Consider two coordinate systems
with origin the central point about which the carousel rotates. The rst
system, X = (x1; x2), is stationary and is oriented in the usual East-West,
North-South fashion, but the second system is attached to the carousel, which
is rotating counterclockwise with constant angular speed ! with respect to
X. The rotating system is denoted by Yt = (yt1; yt2), and at 1:00 p.m. t = 0
and Yt = Y0 = X.
(a) Find a matrix O such that X(p)T = OYt(p)T for each point p in space.

O = (cos(tw) -sin(tw)
sin(tw) cos(tw))

(b) Find the position of the train as a function of t as measured by both X
and Yt:

So I know that the train moves along the direction vector (cos(pi), sin(pi))
so... positionx(t) = (0, -2) + tv(1,0) = (tv, -2)
then does positiony(t) = (cos(tw) -sin(tw) (tv
sin(tw) cos(tw)) -2)

(c) Find the velocity of the train in both systems.

I could do this part if I had both positions (stationary and moving one), but not confident in part (b)
(d) Let Z be a coordinate system attached to the train oriented in the usual
north-south, east-west fashion. Find the relation linking Z coordinates to
those of X:

(d) Which of the three coordinate systems are inertial? Why?

I know how to determine if X, Y, Z are inertial, but not sure what X, Y, and Z are.

(e) A bird is spotted at X = (0; 0)
ying with speed u on a course 45o east of
due north as viewed by the stationary observer X: Find its position vector
relative to X;Y; and Z:

#### tkhunny

##### Moderator
Staff member
That's a lot to read. I found myself hoping you would demonstrate YOUR work or YOUR thoughts or impressions. That IS where we need to start.

#### Subhotosh Khan

##### Super Moderator
Staff member
A train moves due West at a constant speed v. It passes a carousel two
mile due South of the carousel at 1:00 p.m. Consider two coordinate systems
with origin the central point about which the carousel rotates. The rst
system, X = (x1; x2), is stationary and is oriented in the usual East-West,
North-South fashion, but the second system is attached to the carousel, which
is rotating counterclockwise with constant angular speed ! with respect to
X. The rotating system is denoted by Yt = (yt1; yt2), and at 1:00 p.m. t = 0
and Yt = Y0 = X.
(a) Find a matrix O such that X(p)T = OYt(p)T for each point p in space.

O = (cos(tw) -sin(tw)
sin(tw) cos(tw))

(b) Find the position of the train as a function of t as measured by both X
and Yt:

So I know that the train moves along the direction vector (cos(pi), sin(pi))
so... positionx(t) = (0, -2) + tv(1,0) = (tv, -2)
then does positiony(t) = (cos(tw) -sin(tw) (tv
sin(tw) cos(tw)) -2)

(c) Find the velocity of the train in both systems.

I could do this part if I had both positions (stationary and moving one), but not confident in part (b)
(d) Let Z be a coordinate system attached to the train oriented in the usual
north-south, east-west fashion. Find the relation linking Z coordinates to
those of X:

(d) Which of the three coordinate systems are inertial? Why?

I know how to determine if X, Y, Z are inertial, but not sure what X, Y, and Z are.

(e) A bird is spotted at X = (0; 0)
ying with speed u on a course 45o east of
due north as viewed by the stationary observer X: Find its position vector
relative to X;Y; and Z:
An inertial co-ordinate system has NON-ACCELERATING origin.