Question

It is known from the newspapers that the cosmonaut Leonov, going for a walk in space, threw the lens cap of his movie camera toward the earth. Where did it go?

Short answer

Answer: The possible paths of the lens cap after being thrown by cosmonaut Leonov are (1) it falls back to Earth, or (2) it goes into an elliptical orbit around the Earth. The path taken by the lens cap depends on the initial velocity, angle, and direction of the throw. Other factors such as air resistance and gravitational forces also affect the trajectory and final location of the lens cap.

Step-by-step solution

Assumptions and simplifications

To solve this problem, we'll make the following assumptions: 1. The effect of air resistance is ignored. 2. Cosmonaut Leonov, the lens cap, and the Earth are considered as point masses. 3. The only force acting on the lens cap is the gravitational attraction between the Earth and the lens cap. In reality, the problem would be more complex, but these simplifications allow us to demonstrate the concepts involved in an easy-to-understand manner.

Orbital velocity of the lens cap

Before the lens cap is thrown, it has the same orbital velocity as cosmonaut Leonov. Orbital velocity is the minimum velocity an object must have in order to remain in a stable orbit around the Earth, ignoring air resistance. Since the lens cap is thrown towards Earth, it will reduce its orbital velocity.

Gravitational force on the lens cap

The force acting on the lens cap because of the gravitational attraction between the lens cap and the Earth can be given by the formula: F = G * (m1 * m2) / r^2, where: - F is the force acting on the lens cap, - G is the gravitational constant (approximately 6.674 * 10^{-11} N(m/kg)^2), - m1 is the mass of Earth (approximately 5.972 * 10^{24} kg), - m2 is the mass of the lens cap, - r is the distance between the Earth's center and the lens cap. This attractive gravitational force will change the motion of the lens cap.

Analyze the path of the lens cap

Since the lens cap is thrown towards Earth, its trajectory will be affected by the Earth's gravity. Instead of following a straight path, the lens cap will follow a curved path that might intersect the Earth's surface, causing it to fall to the ground. If the lens cap doesn't intersect the Earth's surface, it will continue orbiting the Earth in an elliptical orbit. Hence, depending on the velocity and angle at which the lens cap was thrown, it can follow one of the following paths: 1. It falls back to Earth. 2. It goes into an elliptical orbit around the Earth.

Final destination of the lens cap

The final destination of the lens cap will depend on the path it takes. If it falls back to Earth, it can land anywhere on the surface depending on the initial velocity, angle, and direction of the throw. If the lens cap goes into an elliptical orbit, it will continue to orbit the Earth and might collide with other space debris or burn up upon re-entry. In conclusion, this is a simplified analysis of a real-world scenario and ignores other variables that would impact the lens cap's final location. The actual trajectory and final location would be based on a detailed analysis using more accurate initial conditions, including the effects of air resistance and other forces.

Key concepts

Gravitational Force

Gravitational force is the attractive force between two masses. It is what keeps us tethered to the Earth and plays a crucial role in the movement of celestial objects. This force is governed by Newton's Law of Universal Gravitation, which states that the force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them. Mathematically, it is expressed as:
\[F = G \cdot \frac{m_1 \cdot m_2}{r^2},\]where:

• \(F\) is the gravitational force between the two masses,
• \(G\) is the gravitational constant (approximately \(6.674 \times 10^{-11} \, \text{N(m/kg)}^2\)),
• \(m_1\) and \(m_2\) are the masses of the two objects,
• \(r\) is the distance between the centers of the two masses.

Understanding gravitational force helps explain why the lens cap, once released by Leonov, continued to be pulled towards Earth. This relentless pull dictates the trajectory of the cap, possibly pulling it into an orbit or causing it to fall to Earth.

Orbital Velocity

Orbital velocity is the speed an object must maintain to stay in orbit around a celestial body like Earth. This velocity ensures that the object doesn't fall into the body it is orbiting due to gravitational pull. For any satellite or an object in orbit, like the lens cap in this scenario, the orbital velocity is crucial to its path and behavior in space.
The formula for calculating orbital velocity is given by:
\[V = \sqrt{\frac{G \cdot M}{r}},\]where:

• \(V\) is the orbital velocity,
• \(G\) is the gravitational constant,
• \(M\) is the mass of the celestial body being orbited (for example, Earth),
• \(r\) is the distance from the center of the mass to the orbiting object.

For Leonov's lens cap, initially moving with the same orbital velocity as the spacecraft, any deviation in this speed — such as being thrown towards the Earth — would alter its course. If its speed drops below this threshold, it might fall back to Earth unless it retains enough velocity for an altered, perhaps elliptical, orbit.

Elliptical Orbit

An elliptical orbit is the path taken by an object that is moving along a closed, elongated curve around a celestial body. Unlike a circular orbit where the distance from the object to the central body remains constant, in an elliptical orbit, this distance changes as the object moves. This is described by Kepler's First Law of Planetary Motion, which states that planets move in elliptical orbits with the Sun at one focus.
In the context of the thrown lens cap, if it attains sufficient velocity and the correct angle, it could enter into an elliptical orbit around Earth. A few key characteristics of such an orbit include:

• The orbit's shape is defined by its eccentricity; a value of 0 describes a perfect circle, while values close to 1 indicate a more stretched elongation.
• The farthest point in the orbit from Earth is known as the apogee, while the closest point is the perigee.
• The speed of the object varies: fastest at perigee and slowest at apogee due to the variation in gravitational pull.

Understanding elliptical orbits provides insight into how varying speeds and directions impact an object's journey in space, helping to predict potential outcomes for objects like the lens cap when they are influenced by gravitational forces.