Question

An $80kg$astronaut has gone outside his space capsule to do some repair work. Unfortunately, he forgot to lock his safety tether in place, and he has drifted $5.0m$away from the capsule. Fortunately, he has a $1000W$portable laser with fresh batteries that will operate it for $1.0h$. His only chance is to accelerate himself toward the space capsule by firing the laser in the opposite direction. He has a role="math" localid="1649937737036" $10-h$ supply of oxygen. How long will it take him to reach safety?

Short answer

The remaining distance for the Astronaut to travel is$t=31.533\mathrm{s}\to 8.76\mathrm{h}$.

Step-by-step solution

Definition of distance travel:

The distance travelled by a moving object in a given time interval is the length of the object's actual path between its initial and final positions.

Given values:

$$\begin{gathered} c=3\times {10}^{8}m/s \\ m=80kg \\ P=1000W \\ t=1.0hour=3600s \\ d=5.0m \end{gathered}$$

Substitute the equation:

After the first hour, the astronaut is traveling the speed of:

$v=\frac{P·t}{m·c}$

$v=\frac{1000\mathrm{W}\times 3600\mathrm{s}}{80\mathrm{kg}\times \left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}$

$v=1.5\times {10}^{-4}m/s$

The distance of Astronauts travelled is: 

Astronaut has travelled a distance $x$:

$x=\frac{P}{2mc}{t}^2$

$x=\frac{1000\mathrm{W}}{2(80\mathrm{kg})\left(3\times {10}^8\mathrm{m}/\mathrm{s}\right)}(3600\mathrm{s}{)}^2$

role="math" localid="1649938693985" $x=0.270m$

Finally, to find the travel remaining distance:

Where $△x$is Distance and $v$is Speed:

$t=\frac{\Delta x}{v}$

$t=\frac{5.0\mathrm{m}-0.270\mathrm{m}}{1.5\times {10}^{-4}\mathrm{m}/\mathrm{s}}$

$t=\frac{4.73\mathrm{m}}{1.5\times {10}^{-4}\mathrm{m}/\mathrm{s}}$

$t=31.533\mathrm{s}\to 8.76\mathrm{h}$

Final result:

The result of astronauts remaining distance to travel is $t=31.533\mathrm{s}\to 8.76\mathrm{h}$.