When a spacecraft travels from Earth to the Moon, the gravitational force from Earth initially opposes this journey. Eventually, the spacecraft reaches a point where the Moon's gravitational attraction overcomes the Earth's gravity. How far from Earth must the spacecraft be for the gravitational forces from the Moon and Earth to just cancel?
A hockey puck is hit on a frozen lake and starts moving with a speed of 14.4 m/s. 3.0 seconds later, its speed is 7.3 m/s.
(a) What is its average acceleration?
To determine the distance from Earth where the gravitational forces from the Moon and Earth cancel out, we need to find the point where the gravitational forces from the Moon and Earth are equal in magnitude.
The gravitational force between two objects can be calculated using Newton's law of universal gravitation, which states that the force (F) between two objects is given by:
F = G * (m1 * m2) / r^2
Where:
G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2 kg^-2)
m1 and m2 are the masses of the two objects (Earth and the spacecraft, in this case)
r is the distance between the centers of the two objects
Since we want the gravitational forces from the Moon and Earth to cancel each other out, their magnitudes must be equal. Therefore, we can set up the following equation:
G * (m1 * m2) / r^2 (Moon) = G * (m1 * m2) / r^2 (Earth)
The masses of the spacecraft (m1) and the Moon (m2) cancel out, giving:
r^2 (Earth) = r^2 (Moon)
Simplifying this equation, we get:
r (Earth) = r (Moon)
So, the distance from Earth where the gravitational forces from the Moon and Earth cancel each other out is equal to the distance from the Moon. In other words, the spacecraft must be at the same distance from Earth as the Moon to experience the cancellation of gravitational forces. The average distance from Earth to the Moon is approximately 384,400 kilometers (238,900 miles). Therefore, the spacecraft must be around 384,400 kilometers or 238,900 miles from Earth for the gravitational forces to just cancel out.