Question

Even though the Moon does not have an atmosphere, the trajectory of a projectile near its surface is only approximately a parabola. This is because the acceleration due to gravity near the surface of the Moon is only approximately constant. Describe as precisely as you can the actual shape of a projectile's path on the Moon, even one that travels a long distance over the surface of the Moon.

Short answer

In summary, the actual shape of a projectile's path on the Moon is a complex curve that deviates from an ideal parabola or ellipse. This is due to the Moon's varying gravitational acceleration and the absence of an atmosphere. Over short distances, the path is approximately parabolic, but over longer distances, the varying gravity causes the path to deviate from this shape. The projectile's path can be roughly described as a series of parabolic segments connected by points where the projectile collides with the Moon's surface, ultimately forming a complex curve.

Step-by-step solution

Projectile motion on Earth

On Earth, the path of a projectile is parabolic due to the constant acceleration due to gravity and the absence of any external force (ignoring air resistance). This means that the projectile follows a quadratic equation in terms of its horizontal and vertical positions, with time as the independent variable.

Gravity on the Moon

The Moon's gravity is not constant due to its non-uniform density and irregular shape. This means that the acceleration due to gravity experienced by a projectile on the Moon varies with its position. However, this variation is relatively small over short distances (a few kilometers) and can be considered approximately constant.

Atmospheric effects

The Moon has no atmosphere, which means that there are no external forces such as air resistance acting on the projectile, which simplifies our analysis.

The actual shape of a projectile's path on the Moon

On the Moon, a projectile's path will be affected by both the varying gravitational acceleration and the lack of atmosphere. Over short distances, the variation in gravity can be ignored, and the path will be approximately parabolic. However, over longer distances, the varying gravitational acceleration will affect the projectile's path, causing it to deviate from the parabolic shape. The actual shape of a projectile's path on the Moon will be an ellipse, with the Moon's center of mass acting as a focus for this ellipse. This is because the Moon's gravitational field is the central force acting on the projectile, causing it to move in an elliptical orbit around the Moon's center of mass. However, this elliptical path is only valid if the projectile doesn't collide with the Moon's surface. If the projectile travels a long distance over the surface of the Moon, its actual path will be a series of parabolic segments (due to the local approximation of constant gravity) that are connected by points where the projectile collides with the surface. These segments combine to form a complex curve that deviates from an ideal parabola or ellipse.