"Tom Roberts" escreveu na mensagem
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El Enrrabadore-mor wrote:
The only thing I can guarantee is that the derivative
of the above solution of the integral, gives the function
used by Tom Roberts in first place.[...]
The right solution must be:
\tau = 1/2*(v(t)/c)*sqrt(1-v(t)^2/c^2) + 1/2*arcsin(v(t)/c)

This cannot possibly be correct. Differentiate your expression by t and
you must get a factor of v'(t) (i.e. dv(t)/dt), but there is no such
factor in the original integral.

One thing is for sure. The derivative of the solution must
give the original function (inside the integral).

So let's do the derivative of your solution (and mine)
and see who got the correct solution.

1 - Your solution:
\tau = sqrt(1-|v|^2/c^2)

2 - My solution:
\tau = 1/2*(v/c)*sqrt(1-v^2/c^2) + 1/2*arcsin(v/c)

The original integral:
\tau = \integral sqrt(1-v(t)^2/c^2) dt

If you are to be correct, then:
d/dt [sqrt(1-|v|^2/c^2)] = sqrt(1-|v|^2/c^2)
which obviously looks wrong.
(the derivative of the function equal to the function
itself sounds automatically bad).

The derivative of your solution is:
d/dt [sqrt(1-|v|^2/c^2)] = d/dt [sqrt(1-u^2)]
being u = v/c
and du/dt = (1/c) dv/dt
d/dt [sqrt(1-u^2)] = d/dt [(1-u^2)^1/2] =
= 1/2 (1-u^2)^-1/2 * du/dt =
= 1/2 (1/sqrt(1-u^2)) * (1/c) dv/dt =
= 1/(2*sqrt(1-u^2) * (1/c) dv/dt
For a constant velocity v:
d/dt sqrt(1-u^2) = 1/[2c*sqrt(1-(v/c)^2]

Now, the derivative of my solution is a little
bit hard.
d/dt [1/2*(v/c)*sqrt(1-v^2/c^2) + 1/2*arcsin(v/c)]
being:
u = v/c
w = sqrt(1-v^2/c^2)
z = arcsin(v/c) = arcsin(u)

Hence:
1/2 * d/dt [uw + z] = 1/2 [u dw/dt + w du/dt + dz/dt]

du/dt = (1/c) dv/dt

dw/dt = 1/(2*sqrt(1-u^2) * (1/c) dv/dt =
= 1/[2c*sqrt(1-(v/c)^2]*dv/dt

dz/dt = d/dt [arcsin(u)] = 1/(sqrt(1-u^2) * du/dt =
= 1/[sqrt(1-(v/c)^2]*(1/c)dv/dt

Hence:
1/2 d/dt [uw + z] = 1/2 [(v/c)*1/[2c*sqrt(1-(v/c)^2]*dv/dt +
+ sqrt(1-v^2/c^2)*(1/c)*dv/dt + 1/[sqrt(1-(v/c)^2]*(1/c)dv/dt] =

= 1/2 (1/c) dv/dt [ v/[2c*sqrt(1-(v/c)^2] +
+ sqrt(1-(v/c)^2) + 1/[sqrt(1-(v/c)^2] ]

Now, multiplying by 1 = sqrt(1-u^2)/sqrt(1-u^2) to take
out the denominator:
= 1/2 dv/dt (1/c) (1/sqrt(1-(v/c)^2) [v/2c + 1 - (v/c)^2 + 1]
= 1/2 dv/dt (1/c) (1/sqrt(1-(v/c)^2) [2 + v/2c - (v/c)^2]
Damn, ...
Can't see what's wrong.

Any help, please?
Yes, I do see the predicted dv/dt term.


Tom Roberts