Question

A rocket is accelerated to a speed of \(v=2 \sqrt{g R_{\mathrm{E}}}\) near the Earth's surface and then coasts upward. (a) Show that it will escape from the Earth. ( \(b\) ) Show that very far from the Earth its speed is \(v=\sqrt{2 g R_{\mathrm{E}}}\).

Short answer

The rocket will escape from the Earth as its speed is greater than Earth's escape velocity. The speed of the rocket far away from the Earth will be \(v=\sqrt{2 g R_{\mathrm{E}}}\).

Step-by-step solution

Understanding the Escape Velocity Concept

The escape velocity is the minimum velocity an object must attain to escape the gravitational pull of a planet or other body. The formula of escape velocity is \(v_e = \sqrt{2gR_{\mathrm{E}}}\), where \(g\) is the acceleration due to gravity and \(R_{\mathrm{E}}\) is the Earth's radius.

Compare Given Velocity with Escape Velocity

The rocket's velocity is given as \(v=2 \sqrt{g R_{\mathrm{E}}}\). Compare it with the escape velocity formula. It can be observed that the given velocity is greater than the escape velocity. Therefore, the rocket will escape from the Earth's surface.

Applying the Law of Conservation of Energy

To find the velocity of the rocket far away from the Earth, use the law of conservation of energy. According to this law, the total energy of a system remains constant if no external force acts on it. In the case of the rocket, the total energy includes kinetic and potential energy.

Set Up the Energy Conservation Equation

The total energy at the surface of Earth (when the rocket is just launched) is the sum of kinetic and potential energy, i.e., \(E_1 = \frac{1}{2} m v^2 - \frac{GmM}{R_{\mathrm{E}}}\). Far away from the Earth (when the influence of Earth's gravity is negligible), the total energy is \(E_2 = \frac{1}{2} m v_f^2\), where \(v_f\) is the final velocity of the rocket. Since the total energy is conserved, one can equate \(E_1\) and \(E_2\) and determine \(v_f\).

Calculation of Rocket's Speed Far Away from Earth

Based on the energy conservation equation obtained in Step 4, one can solve for \(v_f\). This results in \(v_f=\sqrt{2 g R_{\mathrm{E}}}\), which finally gives the speed of the rocket far away from the Earth.

Key concepts

Conservation of Energy

The conservation of energy is a crucial principle in physics. It states that the total energy in a closed system remains constant over time. For a rocket leaving Earth, this means that the total energy it has at launch will be the same when it is far away.

In this scenario, we consider two main types of energy: kinetic energy (energy of motion) and gravitational potential energy (energy due to position). By balancing these energies, we can predict how fast the rocket will be moving when it is far away from Earth.

When the rocket is near the Earth, its high speed means high kinetic energy and some gravitational potential energy. By unfaltered conservation, when it’s far away, this transforms into lower kinetic energy as its potential energy approaches zero.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses because of its position in a gravitational field.

For a rocket on Earth, it is expressed as \(U = -\frac{GmM}{R_{\mathrm{E}}}\), where:

• \(G\) is the gravitational constant,
• \(m\) is the mass of the rocket,
• \(M\) is the mass of Earth,
• \(R_{\mathrm{E}}\) is Earth’s radius.

The negative sign indicates that work must be done to move the object against Earth's gravity.

As the rocket rises and moves further from Earth, this potential energy decreases in magnitude, getting closer to zero, because the gravitational pull weakens.

Kinetic Energy

Kinetic energy is the energy an object has due to its motion. It is given by the formula \(K = \frac{1}{2}mv^2\). Higher speeds mean higher kinetic energy.

For a rocket, as it gains altitude, the velocity at which it was launched plays a significant role. Initially, near Earth's surface, our rocket has a great deal of kinetic energy thanks to its high velocity. This initial energy allows it to overcome Earth's gravity.

As it travels away, some of this kinetic energy is used to compensate for the pull of gravity, reducing its speed but allowing it to eventually escape Earth's influence with a residual velocity.

Rocket Motion

Rocket motion is primarily about getting a rocket to achieve enough speed to leave Earth's gravitational pull.

For a rocket to escape Earth, it must reach escape velocity, which is the speed needed to break free from gravitational attraction.

In our exercise, the rocket starts at a speed of \(v=2 \sqrt{g R_{\mathrm{E}}}\). This speed ensures it can overcome gravitational forces. As it moves, its trajectory and speed can be analyzed by considering conservation principles.

Far from the Earth, when gravitational effects are minimal, the rocket will travel at a final speed of \(v_f=\sqrt{2 g R_{\mathrm{E}}}\), showing how initial energy converts into final kinetic form.