Question

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.

Short answer

It is possible to outrun a light ray.

Step-by-step solution

Write the given data from the question.

The force generated by the force is constant.

The initial velocity is constant.

Show that it is possible to outrun a light ray.

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.