Question

answer 3.123×10⁷
problem set 3:
two astronauts find themselves at rest in space, separated by a distance of 5 meters. the mass of the astronauts are 65 kg and 80 kg.

1. determine the strength of the gravitational force acting on the astronauts.

starting equation
$f_g=g\frac{m_1m_2}{r^2}$
substitution
$f_g=g\frac{65×80}{25}$
answer
$1.38736×10^{-8}$

1. determine the acceleration of the astronauts.

starting equation
$f_{net}=ma$ $a=f_{net}/m$
substitution
$a=f_{net}/65$ $a=f_{net}/80$
$65kg=astro-1$
$80kg=astro-2$
answer
$a_{stro-1}=2.1344×10^{-10}$ $a_{stro-2}=1.7342×10^{-10}$

1. if they accelerated constantly at the rate above, how far away from each other will they be after 1 year?

starting equation
$\delta x=\frac{1}{2}at^2+v_it$
substitution
$t=31536000s$
$v_i=0$
$\delta x=\frac{1}{2}at^2$
$106135.0993$ $86234.76816$
answer
$192374.8675m$

Explanation:

Step1: Verify gravitational force

Starting equation: $F_G = G\frac{m_1m_2}{r^2}$
$G=6.674\times10^{-11}\frac{N\cdot m^2}{kg^2}$, $m_1=65kg$, $m_2=80kg$, $r=5m$
$F_G = 6.674\times10^{-11}\times\frac{65\times80}{5^2}=6.674\times10^{-11}\times\frac{5200}{25}=6.674\times10^{-11}\times208=1.388\times10^{-8}N$

Step2: Verify accelerations

For 65kg astronaut: $a_1=\frac{F_G}{m_1}=\frac{1.388\times10^{-8}}{65}=2.135\times10^{-10}\frac{m}{s^2}$
For 80kg astronaut: $a_2=\frac{F_G}{m_2}=\frac{1.388\times10^{-8}}{80}=1.735\times10^{-10}\frac{m}{s^2}$

Step3: Calculate total distance moved

First, convert 1 year to seconds: $t=365\times24\times3600=31536000s$
Distance for astronaut 1: $\Delta x_1=\frac{1}{2}a_1t^2=\frac{1}{2}\times2.135\times10^{-10}\times(31536000)^2$
$\Delta x_1=1.0675\times10^{-10}\times9.945\times10^{14}=106135.09m$
Distance for astronaut 2: $\Delta x_2=\frac{1}{2}a_2t^2=\frac{1}{2}\times1.735\times10^{-10}\times(31536000)^2$
$\Delta x_2=8.675\times10^{-11}\times9.945\times10^{14}=86234.78m$
Total distance apart: $5 + \Delta x_1 + \Delta x_2$

Step4: Compute final separation

$5 + 106135.09 + 86234.78=192374.87m$

Answer:

1. $1.388\times10^{-8}\text{ N}$
2. $2.135\times10^{-10}\frac{m}{s^2}$ (65 kg astronaut), $1.735\times10^{-10}\frac{m}{s^2}$ (80 kg astronaut)
3. $192374.88\text{ m}$