In this post we will consider the velocity function of a space ship, taking relativistic mass and time dilation into account. So suppose we are given such a space ship. While stationary, let its mass be and its power output . Suppose also, that it launches at . Now, its relativistic mass is

and its power output, due to time dilation, is

Since

we have that

Rewriting to conventional differential equation notation we obtain

which has solution

Now let us verify the validity of the solution to the differential equation:

as expected. Now let us calculate the integration constant . Since , we have

and we have that

Now, let us check that all pieces fit together. We have that

Also,

as expected.

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