1-D Relativistic Dynamics

An alternate solution from that provided by @Steven Chase.

Newton's second law reads:

$\vec{F} = \frac{d\vec{p}}{dt} \ ; \ \vec{p} = \frac{mc\vec{V}}{\sqrt{c^2 - \lvert \vec{V} \rvert^2}}$

Applying the above to the given scenario results in:

$\frac{d}{dt}\left(\frac{v}{\sqrt{1-v^2}}\right) = 1$

Since: $v(0) = 0.1$ , solving the above differential equation gives:

$\frac{v}{\sqrt{1-v^2}} - \frac{0.1}{\sqrt{1-(0.1)^2}} = t$

Solving for $t$ when $v=0.9$ gives the result:

$\boxed{t = \frac{9}{\sqrt{19}}-\frac{1}{\sqrt{99}} \approx 1.96425}$

The reason I asked the bonus question was to show that in the non-relativistic case, the particle's speed exceeds that of light, which in reality is impossible.

Start with the relativistic momentum:

$p = \frac{m c v}{\sqrt{c^2 - v^2}}$

Relativistic Newton's 2nd Law (there are no vectors, since the problem is one dimensional):

$F = \frac{d}{dt} p \\ F = \frac{\sqrt{c^2 - v^2} \, m c \, \dot{v} - m c v \, \frac{1}{2} (c^2 - v^2)^{-1/2} \, (-2 v \dot{v})}{c^2 - v^2}$

Simplifying results in:

$F = (\alpha + \beta) \dot{v} \\ \alpha = \frac{mc}{\sqrt{c^2 - v^2}} \\ \beta = \frac{m c v^2}{(c^2 - v^2)^{3/2}}$

The rest is just number crunching. In the non-relativistic case, the acceleration would be $1$ and the particle would reach $v = 0.9 c$ at $t = 0.8$ . It takes considerably longer in the relativistic case because relativistic particles behave as if they have mass greater than their rest mass.

I also tested this with my general $3D$ relativistic mechanics solver, and the results were the same.

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import math

m = 1.0
v0 = 0.1
F = 1.0
c = 1.0
dt = 10.0**(-6.0)

################################

t = 0.0

v = v0
vd = 0.0

while v <= 0.9:

    v = v + vd*dt

    alpha = m*c/math.sqrt(c**2.0-v**2.0)
    beta = m*c*(v**2.0)/((c**2.0-v**2.0)**(3.0/2.0))

    vd = F/(alpha+beta)

    t = t + dt

################################

print dt
print t

#>>> 
#0.0001
#1.9643
#>>> ================================ RESTART ================================
#>>> 
#1e-05
#1.96424
#>>> ================================ RESTART ================================
#>>> 
#1e-06
#1.96423799993
#>>>