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- Summary
- Why does a constant velocity have to be transformed when dealing with rotating reference frames?

Consider a rotating disk with the center at the origin of a stationary Cartesian coordinate system, (x, y).

At t = 0, on the circumference of the disk, someone/something shoots a particle with constant velocity components Dvx, Dvy (where the D indicates the rotating disk). Also at time t=0, the disk coordinate of the shooter lines up with the Cartesian (R, 0) coordinate.

Thus, the initial points are the same for both frames:

(x0, y0) = (Dx0, Dy0) = (R, 0).

Also the initial, and constant, velocity components are equal in both frames:

[vx0, vy0] = [Dvx0, Dxy0].

In the Cartesian frame, the particle trajectory is a straight line:

x = x0 + vx0 * t

y = y0 + vy0 * t

However, if I transform this Cartesian (or inertial) trajectory into the rotating frame using the matrix:

| cos(w*t) sin(w*t) |

| -sin(w*t) cos(w*t) |

where w is the angular rotation velocity, I do not get the expected trajectory as seen in the rotating frame. I get a curved trajectory but the trajectory is not what is should be.

The initial velocity has to also be transformed, but since the initial velocity is the same in both frames, and since the velocity is constant, I don't understand intuitively why this is so.

At t = 0, on the circumference of the disk, someone/something shoots a particle with constant velocity components Dvx, Dvy (where the D indicates the rotating disk). Also at time t=0, the disk coordinate of the shooter lines up with the Cartesian (R, 0) coordinate.

Thus, the initial points are the same for both frames:

(x0, y0) = (Dx0, Dy0) = (R, 0).

Also the initial, and constant, velocity components are equal in both frames:

[vx0, vy0] = [Dvx0, Dxy0].

In the Cartesian frame, the particle trajectory is a straight line:

x = x0 + vx0 * t

y = y0 + vy0 * t

However, if I transform this Cartesian (or inertial) trajectory into the rotating frame using the matrix:

| cos(w*t) sin(w*t) |

| -sin(w*t) cos(w*t) |

where w is the angular rotation velocity, I do not get the expected trajectory as seen in the rotating frame. I get a curved trajectory but the trajectory is not what is should be.

The initial velocity has to also be transformed, but since the initial velocity is the same in both frames, and since the velocity is constant, I don't understand intuitively why this is so.