65 starships fly along straight line, the relative speed of two consecutive starships is c/3?

Question

What is relative speed of the first starship with respect to the last starship?

Answer 1

Apply the relativistic velocity addition formula recursively 64 times to get, where v = velocity of last starship:

(((2^64-1) c + (2^64+1) v) / ((2^64+1) c + (2^64-1) v)) c

Then equate this with (v + x) / (1 + (v x)/c^2) and solve for x, getting:

x = ((2^64-1) / (2^64+1)) c

Answer 2

The "easy" way to do this is to recognize that the relativistic speed addition formula is the same one as for the hyperbolic tangent = tanh(x) = (e^x - e^(-x))/(e^x + e^(-x)). That is to say,
tanh(x + y) = (tanh(x) + tanh(y))/(1 - tanh(x)*tanh(y))

What that means is that if you have a speed v-a and a speed v-b, if you find the values of x and y such that:
v-a= c * tanh(x) and
v-b = c * tanh(y), then the speed of the relativistic "sum" of them is
v-sum = c * tanh(x + y).

I won't prove that here, I'll just give a reference below.

How does that help? A lot! Here's 65 ships in a row:
v-0 = 0 (in its own frame, of course)

v-1 = c/3 . This means that tanh(x) = 1/3, or x = ATANH(1/3)
Well, you can use Excel to find out that that means
x = 0.34657. So v-1 = c * tanh(x)

v-2 = c * tanh(2x)
v-3 = c * tanh(3x)
....
v-64 = c * tanh(64x). This is the leading ship.
So what is tanh(64x)?
64x = 64 * 0.34657 = 22.18074
tanh(22.18074) = 1, according to Excel. But we want to know how close to 1 it is. We can do so as follows:
Since tanh(y) = (e^y - e^(-y))/(e^y + e^(-y))
1 - tanh(y) = 2*e^(-y)/(e^y + e^(-y)), which is extremely close to:
2e^(-y)/e^y = 2e^(-2y)

so tanh(y) = 1 - 2e^(-2y)
= 1 - 2(5.42e-20)
= 1 - 1.084e-19
so the actual speed is c*tanh(y)
= c - c*1.084e-19
= c - (2.9979e8)(1.084e-19)
= c - 3.25e-11 m/s

So the difference between the speed of the lead ship and the speed of light is about 3 atomic radii per second.

Answer 3

between 1 and 2 is c/3 = (c/3)^1
between 1 and 3 is (c/3)(c/3) = (c/3)^2
....
between 1 and 65 = (c/3)^64

Answer 4

(c/3)^64.