The mass of the spaceship is $m = 2.5 times 10^{7} mathrm{~kg}$.The distance between the two objects is $d = 2 mathrm{~km}$ from point $B$. The mass of each asteroid is $M = 3.5 times 10^{11} mathrm{~kg}$.

$textbf{Required: }$

Finding the magnitude gravitational forces $F_{1}$ and $F_{2}$ that acting on the spaceship.

As Newton’s law of universal gravitation, The force of gravity between two objects is given by

$$
begin{align*}
F &= G ~ dfrac{ m_{1} ~ m_{2} }{ r^{2} } \
&= G ~ dfrac{ m ~ M }{ r^{2} } \
end{align*}
$$

Since the symmetry in the graph. Then the two tringles are identical. According to Pythagorean’s theorem, the distance $r$ from the point $B$:

$$
begin{align*}
r^{2} &= X^{2} + Y^{2} \
r &= sqrt{ X^{2} + Y^{2} } \
&= sqrt{ left( 2 mathrm{~kg} right)^{2} + left( 1.5 mathrm{~kg} right)^{2} } \
&= 2.5 mathrm{~km}
end{align*}
$$

Substituting from the previous calculations to find the magnitude gravitational forces $F_{1}$ and $F_{2}$ that acting on the spaceship.

$$
begin{align*}
F_{1} &= F_{2} \
&= G ~ dfrac{ m ~M }{ r^{2} } \
&= 6.67 times 10^{-11} mathrm{~N cdot m^{2}/kg^{2}} ~ dfrac{ 2.5 times 10^{7} mathrm{~kg} times 3.5 times 10^{11} mathrm{~kg} }{ left( 2.5 times 10^{3} mathrm{~m} right)^{2} } \
&= 93.38 mathrm{~N}
end{align*}
$$

So, the magnitude gravitational forces $F_{1}$ and $F_{2}$ that acting on the spaceship are $F_{1} = F_{2} = 93.38 mathrm{~N}$.

The mass of the Ceres is $m_{1} = 8.7 times 10^{20} mathrm{~kg}$.The distance between the Ceres and the spaceship is $r = 1.4 times 10^{4} mathrm{~km}$. The mass of the astronaut is $m_{2} = 95 mathrm{~kg}$.

$textbf{Required: }$

Finding the magnitude gravitational force.

Solve for the two cars are identical $m_{1} = m_{2}$:

As Newton’s law of universal gravitation, The force of gravity between two objects is given by

$$
begin{align*}
F &= G ~ dfrac{ m_{1} ~ m_{2} }{ r^{2} } \
&= 6.67 times 10^{-11} mathrm{~N cdot m^{2}/kg^{2}} ~ dfrac{ 8.7 times 10^{20} mathrm{~kg} times 95 mathrm{~kg} }{ left( 1.4 times 10^{7} mathrm{~m} right)^{2} } \
&= 0.028 mathrm{~N}
end{align*}
$$

So, the magnitude of the gravitational force is $0.028 mathrm{~N}$.