State Newton's law of universal gravitation - Leaving Cert Physics - Question 6 - 2010

To find the acceleration due to gravity at a height $h = 2R$ (where $R$ is the radius of the Earth), we use: $g_h = G \frac{M}{(R+h)^2}$ Substituting $h = 2R$, we get: $g_h = G \frac{M}{(R+2R)^2} = G \frac{M}{(3R)^2} = \frac{g}{9}$ Thus, the acceleration due to gravity at this height is $g_h = \frac{9.81}{9} \approx 1.09 \, m/s^2$.

The spacecraft continues on its journey due to Newton's first law of motion, which states that an object in motion will remain in motion unless acted upon by an external force. Once the engines are turned off, there are no significant forces acting on the spacecraft, allowing it to maintain its inertial motion towards the moon.

As the astronauts travel towards the moon, the gravitational pull from the Earth decreases. This is because the gravitational force is inversely proportional to the square of the distance from the center of the Earth. As they move further away, the weight, which is the force exerted by gravity on them, decreases, ultimately approaching zero as they near the moon. Thus, the astronauts will feel lighter as they get farther from Earth.

Astronauts experience weightlessness when the gravitational pull from the moon and the earth is equal at a certain distance above the Earth's surface. This can be calculated by setting the gravitational forces equal: $\frac{GM_Em_e}{R_E^2} = \frac{GM_m m_e}{d^2}$ Solving for $d$: $d = R_E \sqrt{\frac{M_E}{M_m}} \approx 3.39 \times 10^6 \, m$ Thus, the height above Earth's surface can be calculated as: $height = d - R_E = 3.39 \times 10^6 - 6.36 \times 10^6 = 3.39 \times 10^6 \, m$.

To find the velocity of the moon, we use the formula: $v = \frac{2 \pi r}{T}$ Where:

• $r = 2.54 \times 10^6 \, m$, is the average distance of the moon,
• $T = 27.3 \times 24 \times 60 \times 60 \, s$ is the time period for one full orbit. Substituting these values: $v \approx \frac{2 \pi \cdot (2.54 \times 10^6)}{27.3 \times 24 \times 60 \times 60} \approx 102.99 \, m/s$.

The moon does not have a substantial atmosphere because its surface gravity is too weak to hold onto atmospheric gases. The gravitational force acting on the moon is insufficient to prevent these gases from escaping into space, which results in the lack of a sustainable atmosphere.