Show that it is possible to outrun a light ray, if you're given a sufficient head start, and your feet generate a constant force.

The force generated by the force is constant.

The initial velocity is constant.

The expression for position as function of time, with the initial velocity is zero and generated force is constant is given as follows.

$\mathbf{x}\left(\mathbf{t}\right)\mathbf{=}\frac{{\mathbf{mc}}^2}{F}\mathbf{[}\sqrt{\mathbf{1}\mathbf{+}{\left(\frac{\mathrm{ft}}{\mathrm{mc}}\right)}^{\mathbf{2}}}\mathbf{-}\mathbf{1}\mathbf{]}\mathbf{+}{\mathbf{v}}_0\mathbf{t}$

Here,$m$ is the mass,$c$ is the speed of the light,$F$ is the generated force,$v_0$ is the initial force and$t$ is the time.

The expression for the position of the photon is given by,

$x_p\left(t\right)=ct$

The time-position graph is shown below.

From the above graph, the photon which starts from$t<0$ ca easily catches the person in the hyperbolic motion but the photon which starts from$t>0$ would not be able to catch up the person in hyperbolic motion.

Therefore, the outrun is possible.

Hence it is possible to outrun a light ray.