What is the escape speed from the earth for an object that is as far from the earth as the orbit of the moon?

Imagine you are at the bottom (floor) of the well and need to escape (rescue). You need energy and strength to climb out of the well. The deeper the well, the more you need to use additional energy to reach the top of the well than you think. If you have the strength to ascend halfway, you will inevitably descend to the bottom. The escape velocity can calculate the precise amount of energy required to reach the well’s top if no energy is left over after walking away.

Escape velocity can be expressed as the specific amount of energy required to exit the gravitational pull of a mass-bearing object. All things have a quantifiable gravitational pull because they have mass.

Escape Velocity

Escape velocity can be elaborated as the minimum amount of velocity with which a body can be pushed vertically upward to escape (exit) earth’s gravitational pull and never revert home.

When you throw something vertically up, it travels to a given height and then falls back to the ground. However, throwing it with a higher initial velocity rises higher before falling back to the ground.

In this manner, if we continue throwing an item with a specific initial velocity, the item will eventually pass through the earth’s gravitational field and escape (exit) from it.

Escape velocity is the minimum initial velocity at which an item can be propelled away from the earth’s gravitational pull.

What is the escape velocity’s mathematical formula?

An object’s escape velocity can be used to determine the escape velocity formula.

The formula for escape velocity using the earth’s mass, radius, and the gravitational constant

Escape Velocity = ve = √(2 G M / R)

where,

G = gravitational universal constant

M = mass of the earth from where the object is thrown

R = radius of the earth

The letter M denotes the planet’s mass in the equation. Planets with greater mass make it more difficult to escape than planets with lower mass. It is because a planet’s gravitational pull becomes stronger the greater the mass of the planet. For instance, watching videos of astronauts landing on the moon appears effortless. It happens because the moon has a far lower mass than the earth, suppressing its gravity.

The radius, or r, in the equation, denotes the distance between both the object seeking to escape and the planet’s center. In other terms, radius measures the gap between a planet’s surface and its center. The planet’s gravitational pull will have a lower impact on an object as it moves away from it. When an object moves sufficiently far, it experiences essentially little attraction. When this occurs, the escape velocity will essentially be 0.

In the equation, G is a constant. It is specifically the gravitational constant described by Newton. For the time being, all you have to understand is that this constant is required for the equation to function. G is roughly equivalent to 6.67 × 10–11 metres3/(kg)(second)2.

Derivation of Escape Velocity

Consider a sphere-shaped planet with a mass of M and a radius of R. From point A on the earth’s surface, a body with mass m is launched.

Let’s join OA and expand it. Think about two sites, P and Q, with distances x and (x + dx) from the earth’s centre, O.

Let’s assume that ve is the minimum velocity needed to leave the earth’s surface. The formula for an object’s kinetic energy with velocity ‘v’ and mass ‘m’ be:

KE = ½ mve2 

When the object is at point P (at a distance x from the earth’s centre O), the gravitational force between the object and the earth’s surface is:

F = GMm/x2

Now, the effort expended to move the object from point P to point Q against the effects of gravity is given by

dW = F dx = GMm/x2 dx

Work completed transporting the object from distance x to distance dx.

We can determine the total work required to move an object from the earth’s surface to infinity while resisting gravity by incorporating the equation from x = R to x = ∞.

Calculation of the overall work performed by an object against the gravitational pull 

So, the total amount of work is:

The kinetic energy imparted must be equivalent to the energy against gravity incurred travelling from the earth’s surface to infinity for the object to depart (escape) from the planet’s surface.

KE = W

Kinetic energy = Work performed against gravity.

Now, let’s replace the value of kinetic energy and work performed in opposition to gravity.

Obtaining the escape velocity by replacing kinetic energy with work against gravity.

Given that we are aware that the acceleration caused by gravity equals

g = GM/R2

ve = 2gR

It provides the escape velocity’s final formula.

The relationship demonstrates that an object’s escape velocity solely depends on the radius and mass of the planet from which it is launched and not on the mass of the projected object.

Escape Velocity of Earth

The formula (mentioned below) can be used to determine the escape velocity of the earth, which is 11.2 km/s:

ve = √ ( 2gR )

The earth’s g (gravitational acceleration) is 9.8 m/s2, and its radius is 6.371 * 106 m. By entering these values in the equation above, we can get the escape velocity of the earth,

The escape velocity of earth = ve 

ve =  √( 2 * 9.8 * 6.371 * 106 )

     =  √( 11.17 * 106 )

     =  11.17 * 103 m/s

  ve of the earth is 11.2 km/s

To escape the earth’s gravitational pull, a spaceship leaving the planet’s surface needs to have an initial velocity of 11.2 km/sec, or 7 miles/sec.

Additionally, because of this substantial value, gas molecules cannot quickly escape from the planet, thus, creating the atmosphere around it.

Escape Velocity of Moon

As discussed above, if you know the radius and mass of any given object, you can compute the escape velocity of the moon. For instance, we may get the moon’s escape velocity using the equation above. The moon has a radius of 1738 kilometres from its equator. Additionally, it is thought to weigh 7.342 1022 kg. The escape velocity of the moon is calculated to be 2.38 km/s. That is far lower than 11.2 km/s required to leave the earth. As a result, gas molecules can easily escape from the moon’s surface. As a result, the moon’s atmosphere is almost absent. Possibly rockets will be constructed on and launched from the moon in the future rather than from earth!

How is something launched into space?

You might be curious about how challenging launching something is, which is so huge! How quickly does it have to move? Interestingly, the same launch speed is needed to send anything from the earth’s surface toward deep space (space beyond the earth’s orbit). It includes the Falcon Heavy, a Roadster, and even a baseball. This speed is known as escape velocity Since it is just fast enough to escape (exit) the earth’s gravitational effect.

But why is the escape velocity constant regardless of the object’s mass? Because (as discussed earlier) mass and escape velocity are unrelated. For instance, you wished to travel 100 kilometres in one hour. It wouldn’t matter if you operated a small automobile or a large freight vehicle. To get there, you would still have to travel at 100 km/h.

Conclusion

Escape velocity is the speed at which a body must move to escape the gravitational pull of a planet or the moon. In the physics field of kinematics, the derivation of escape velocity is a meaningful concept, and questions on it are frequently included in school exams. To fully comprehend the linked concepts and perceive the in-depth notions, the derivation of escape velocity is very crucial. Using the escape velocity formula derivation, you can calculate the minimum velocity an object needs to escape a planet’s or object’s gravitational influence.

Frequently Asked Questions

1. What precisely is the escape velocity from the earth’s surface? 

Ans. It is an incredible 11.2 km/s (kilometers per second). To be exact, it is more than  40,000 km/h. You could get from the North to the South Pole in around 21 minutes at that speed!

2. Which planet has the slowest rate of escape?

Ans. The planet with the lowest mass has the slowest rate of departure in our solar system. Mercury has the slowest escape velocity in the planetary system. Mercury’s escape speed is 4.25 km/s.

3. How is escape velocity determined?

Ans. The velocity required to exit orbit and the gravitational field governing that orbit can be calculated by multiplying the velocity needed to maintain orbit at a particular height and multiplying it by the square root of 2 (roughly 1.414).