The mass of the lunar rover on the Apollo missions is $m = 225 mathrm{~kg}$.

$textbf{Required: }$

Finding the weight of the lunar rover on the Apollo missions on Earth and on the Moon.

Solve for the weight of the lunar rover on the Apollo missions on Earth:

In order to evaluate the weight of the lunar rover on the Apollo missions, we use the following relation:

$$
begin{align*}
W_{E} &= m ~ a_{E} \
&= 225 mathrm{~kg} times 9.8 mathrm{~m/s^{2}} \
&= 2205 mathrm{~N}
end{align*}
$$

So, the weight of the lunar rover on the Apollo missions on Earth is $2205 mathrm{~N}$.

Solve for the weight of the lunar rover on the Apollo missions on Moon:

As we know the accerlation due to the gravity on the Moon is $1.62 mathrm{~m/s^{2}}$. In order to evaluate the weight of the lunar rover on the Apollo missions, we use the following relation:

$$
begin{align*}
W_{M} &= m ~ a_{M} \
&= 225 mathrm{~kg} times 1.62 mathrm{~m/s^{2}} \
&= 364.5 mathrm{~N}
end{align*}
$$

So, the weight of the lunar rover on the Apollo missions on Moon is $364.5 mathrm{~N}$.

The weight of the lunar rover on the Apollo missions on Moon is $364.5 mathrm{~N}$.

The mass of the object is $m_{1} = 4.6 mathrm{~kg}$. The mass of Earth is $m_{2} = 5.97 times 10^{24} mathrm{~kg}$. The gravitational force between the object and the Earth is $F = 2.2 mathrm{~N}$.

$textbf{Required: }$

(a) Finding the distance between the object and the Earth.

(b) Finding the initial acceleration of the object.

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

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

Rearrange and solve for the distance between the object and the Earth:

$$
begin{align*}
r &= sqrt{ G ~ dfrac{ m_{1} ~ m_{2} }{ F} } \
&= sqrt{ 6.67 times 10^{-11} mathrm{~N cdot m^{2}/kg^{2}} ~ dfrac{ 4.6 mathrm{~kg} times 5.97 times 10^{24} mathrm{~kg} }{ 2.2 mathrm{~N} } } \
&= 2.885 times 10^{7} mathrm{~m}
end{align*}
$$

So, the distance between the object and the Earth is $2.885 times 10^{7} mathrm{~m}$.

According to Newton’s second law:

$$
begin{align*}
sum F &= m ~ a \
end{align*}
$$

Rearrange and solve for the initial acceleration of the object:

$$
begin{align*}
a &= dfrac{ F}{m} \
&= dfrac{2.2 mathrm{~N} }{ 4.6 mathrm{~kg} } \
&= 0.478 mathrm{~m/s^{2}}
end{align*}
$$

So, the initial acceleration of the object is $0.478 mathrm{~m/s^{2}}$.

(b) The initial acceleration of the object is $0.478 mathrm{~m/s^{2}}$.