Question

An $80kg$spacewalking astronaut pushes off a $640kg$satellite, exerting a $100N$force for the role="math" localid="1649256566272" $0.50\text{s}$it takes him to straighten his arms. How far apart are the astronaut and the satellite after $1min.$

Short answer

The distance between astronaut and satellite after $1min$islocalid="1650275055209" $2530.8\mathrm{m}.$

Step-by-step solution

Given information

An $80kg$spacewalking astronaut pushes off a $640kg$satellite, exerting a $100N$force for the $0.50s$ it takes him to straighten his arms.

Calculation

The astronaut pushes the satellite with a force of $100\mathrm{N}$and by Newton's $3rd\text{law}$,the force acting on the astronaut is also the same but in opposite direction.

The acceleration of the astronaut (a) can be evaluated by :-

localid="1649257431124" $$\begin{gathered} F=m_{1}a \\ \Rightarrow a=\frac{F}{m_1} \\ \Rightarrow a=\frac{100N}{80kg} \\ \Rightarrow a=1.25\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

Similarly, acceleration of the satellite $\left({\mathrm{a}}_1\right)$can be evaluated by $F=m_2{\mathrm{a}}_1$

localid="1649257555405" $$\begin{gathered} \Rightarrow a_1=\frac{F}{m_2} \\ \Rightarrow a_1=\frac{100N}{640kg} \\ \Rightarrow a_1=0.156\mathrm{m}/{\mathrm{s}}^2\dots \left(2\right) \end{gathered}$$

Continuation of calculation

Relative acceleration $=(.156+1.25)m{s}^{-2}=1.406m{s}^{-2}$

Distance between them after 60 s can be calculated as follows

Distance $$\begin{gathered} s=\frac{1}{2}\times 1.406\times {60}^2 \\ =2530.8m. \end{gathered}$$

Final answer

The distance between astronaut and satellite after $1min$islocalid="1650275142622" $2530.8\mathrm{m}.$