A body is launched from earth with velocity greater than 10% the escape velocity (E). When the body escapes the gravitional pull of the earth, what is the velocity still left with the body

To understand the velocity of a body that has been launched from Earth and escapes its gravitational pull, we first need to clarify a few concepts related to escape velocity and energy conservation. Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body without any additional propulsion. For Earth, this value is approximately 11.2 kilometers per second (km/s). If a body is launched with a velocity greater than 10% of this escape velocity, we can analyze what happens as it escapes Earth's gravity.

Understanding Escape Velocity

Escape velocity (E) is calculated using the formula:

• E = √(2GM/R)

Where:

• G is the gravitational constant (approximately 6.674 × 10^-11 N(m/kg)²),
• M is the mass of the Earth (about 5.972 × 10^24 kg), and
• R is the radius of the Earth (approximately 6.371 × 10^6 meters).

When we say that a body is launched with a velocity greater than 10% of the escape velocity, we are referring to a speed greater than 1.12 km/s (10% of 11.2 km/s). This means the body has enough kinetic energy to overcome the gravitational potential energy of Earth.

Energy Considerations

As the body ascends, it converts kinetic energy into gravitational potential energy. The total mechanical energy (E_total) of the body can be expressed as:

• E_total = Kinetic Energy + Potential Energy

At the surface of the Earth, the kinetic energy (KE) can be calculated using:

• KE = 0.5 * m * v²

Where m is the mass of the body and v is its initial velocity. The potential energy (PE) at the surface is given by:

• PE = -GMm/R

As the body escapes, it will reach a point where its gravitational potential energy becomes zero (at infinite distance from Earth), and all of its initial kinetic energy will be converted into potential energy. However, if the body has more than the minimum escape velocity, it will still retain some kinetic energy as it moves away from Earth.

Calculating Remaining Velocity

Let’s say the body is launched with a velocity of v_initial, which is greater than 1.12 km/s. The total energy at launch can be expressed as:

• E_total_initial = 0.5 * m * v_initial² - GMm/R

When the body escapes, its potential energy at a very large distance is zero, so:

• E_total_final = 0.5 * m * v_final² = E_total_initial

From this, we can derive the final velocity (v_final) when the body has escaped Earth's gravitational influence:

• 0.5 * m * v_final² = 0.5 * m * v_initial² - GMm/R

By simplifying, we can find:

• v_final² = v_initial² - (2GM/R)

Thus, the remaining velocity of the body after escaping Earth's gravity depends on its initial velocity and the gravitational parameters of Earth. If the initial velocity is significantly greater than the escape velocity, the body will still have a substantial speed as it moves away from Earth.

Example Calculation

For instance, if the body is launched with an initial velocity of 12 km/s:

• v_initial = 12,000 m/s
• Using the values for G, M, and R, we can calculate the gravitational potential energy term.

After performing the calculations, you would find that the body retains a final velocity that is still positive, indicating it continues to move away from Earth with some speed.

In summary, a body launched with a velocity greater than 10% of the escape velocity will escape Earth's gravitational influence, but it will still retain some velocity as it moves away, depending on its initial speed and the gravitational energy it overcame. This interplay of kinetic and potential energy is crucial in understanding the motion of objects in space.