Problem 753 - JEE Physics

Question

The mass of a space ship is \(1000 \mathrm{~kg} .\) It is to be launched from earth's surface out into free space the value of \(\mathrm{g}\) and \(\mathrm{R}\) (radius of earth) are \(10 \mathrm{~ms}^{-2}\) and \(6400 \mathrm{~km}\) respectively the required energy for this work will be \(=\ldots \ldots \ldots .\) J (A) \(6.4 \times 10^{11}\) (B) \(6.4 \times 10^{8}\) (C) \(6.4 \times 10^{9}\) (D) \(6.4 \times 10^{10}\)

Step-by-Step Solution

Answer

The energy required to launch the spaceship from Earth's surface into free space is approximately (C) \(6.4 \times 10^9 J\).

Step 1: Convert Earth's radius to meters

Since we are given the Earth's radius in kilometers, we need to convert it to meters. We know that 1 km = 1000 m, so we can easily convert 6400 km to meters: R = 6400 km * 1000 m/km = 6,400,000 m

Step 2: Calculate the Gravitational Potential Energy (GPE)

The formula for calculating the GPE is: GPE = - (G * M * m) / r Where: - GPE is the gravitational potential energy (in joules) - G is the universal gravitational constant (approximately \(6.674 \times 10^{-11} N m^2/kg^2\)) - M is the mass of Earth (approximately \(5.972 \times 10^{24} kg\)) - m is the mass of the spaceship (1000 kg) - r is the distance from the center of Earth (Earth's radius) We already know the values for m and r. Let's plug them into the formula: GPE = - (\(6.674 \times 10^{-11} N m^2/kg^2\) * \(5.972 \times 10^{24} kg\) * 1000 kg) / 6,400,000 m

Step 3: Calculate the energy required to overcome the gravitational force (work)

Now that we've calculated the GPE, to find the energy required, we need to find the work done against the Earth's gravitational force. Since GPE is negative, we need to make the result positive, and also multiply the GPE by the mass of the spaceship to find the energy. Energy required = - GPE Energy required = (\(6.674 \times 10^{-11} N m^2/kg^2\)) * \(5.972 \times 10^{24} kg\) * 1000 kg) / 6,400,000 m Energy required \( \approx 6.4 \times 10^9 J\)

Step 4: Choose the correct answer

Comparing our result with the given options, we find that the energy required to launch the spaceship is approximately: (C) \(6.4 \times 10^9 J\)

Key Concepts

Understanding the Universal Gravitational Constant

In our universe, the universal gravitational constant, often symbolized as \(G\), is a vital component of the law of universal gravitation. \(G\) is crucial because it helps us determine the force of attraction between two masses. This constant has been experimentally found to be approximately \(6.674 \times 10^{-11}~\text{N m}^2/\text{kg}^2\).

Thanks to \(G\):

• We can calculate the gravitational force between two objects based on their masses and the distance between them.
• It provides a universal value that can be used across various scientific calculations, from celestial mechanics to engineering projects.

The importance of \(G\) extends beyond simple calculations. It remains a cornerstone in understanding forces within our galaxy and beyond, making it indispensable in the study of celestial bodies and space physics. Gravitational interactions governed by this constant not only define orbits but also play a role in space-time curvature.

Exploring the Work-Energy Principle

The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In context with gravitational potential energy, it helps us understand how energy is transferred and transformed.

• When work is done against gravity, such as lifting an object, energy storage in the form of gravitational potential energy increases.
• This stored energy can later be converted back into kinetic energy when the object falls, highlighting the interplay between different energy forms.

In our spaceship problem, "work" refers to the energy required to move the spaceship from the Earth's surface to outer space. It is calculated using the gravitational potential energy equation, and it ensures conservation of energy, one of the fundamental principles in physics.

By understanding this principle, students can easily appreciate how energy transactions occur during processes like launching and landing spacecraft, where potential energy is converted to kinetic energy, or vice versa.

Unlocking the Mysteries of Space Physics

Space physics is the study of everything beyond Earth's atmosphere, delving into phenomena like gravitational forces that affect satellites and other celestial bodies. It includes the investigation of:

• The dynamics involved in launching objects into space and controlling their path.
• The impact of celestial mechanics on spacecraft trajectories.
• The role of gravitational potential energy in missions, like keeping satellites in orbit or launching exploration vehicles.

Understanding space physics requires a grasp of concepts like the gravitational force, which dictates how objects interact and move in space. Gravitational potential energy calculations, like those performed in the exercise, are essential for designing and executing successful space missions.

Space physics also connects to the broader science of astrophysics, providing insight into how energy and matter interact on both cosmic and micro scales. This field continues to expand our knowledge about our universe, informing everything from satellite deployment to interstellar exploration.