Question

Short answer

The distance between the astronauts is \(D = 19.76\;{\rm{m}}\).

Step-by-step solution

Centre of mass

Hence, there is only internal force acting on the two astronauts; the centre of mass of the system does not change.

Given data

Given data:

The mass of one of the astronauts is\(m = 55\;{\rm{kg}}\).

The mass of the other astronaut is\(m' = 85\;{\rm{kg}}\).

The distance moved by the lighter astronaut is \(d = 12\;{\rm{m}}\).

Determine the distance parted

The formula of centre of mass is as follows:

\({x_{{\rm{CM}}}} = \frac{{md + m'd'}}{{m + m'}}\)

Here,\(d'\)is the distance moved by the other astronaut.

On plugging the values in the above equation,

\(\begin{array}{l}0 = \left[ {\frac{{\left( {55\;{\rm{kg}}} \right)\left( {12\;{\rm{m}}} \right) + \left( {85\;{\rm{kg}}} \right)d'}}{{\left( {55\;{\rm{kg}}} \right) + \left( {85\;{\rm{kg}}} \right)}}} \right]\\d' = - 7.76\;{\rm{m}}\end{array}\).

The distance part is calculated as follows:

\(\begin{array}{l}D = d - d'\\D = \left( {12\;{\rm{m}}} \right) - \left( { - 7.76\;{\rm{m}}} \right)\\D = 19.76\;{\rm{m}}\end{array}\)

Thus, \(D = 19.76\;{\rm{m}}\) is the distance between the two astronauts.