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Using the following rules:

x = vx * t

y = vy * t + c * t

sqr( vx * vx + vy * vy ) = 1

When the numbers c, x and y are known,

how do I calculate t, vx and vy for the lowest positive value of t?

how do I calculate vx and vy for a given value of t?

x = vx * t

y = vy * t + c * t

sqr( vx * vx + vy * vy ) = 1

When the numbers c, x and y are known,

how do I calculate t, vx and vy for the lowest positive value of t?

how do I calculate vx and vy for a given value of t?

although it is routine to calculate the trajectory based on known initial condition, the answer to the reverse problem may not always be unique or finite.

t=sqrt(x^2 + (y-ct)^2) or t = -sqrt(x^2 + (y-ct)^2)

The units do not check out.

(actually possibly two because there is a square root involved)

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from (1), we can have: vx = x/t

from (3), we can have vy = sqrt(1- vx*vx) = sqrt(1- (x/ t) * (x/t))

considering (2), after a series of transformation:

t=sqrt(x^2 + (y-ct)^2) or t = -sqrt(x^2 + (y-ct)^2)

It is can be inferred that the lowest positive value of t is location, (x,y), dependent. With x = 0, y = ct, t can be 0.