What Determines the Escape Speed from Earth's Gravitational Pull?

[hadronthunder]

Hi friends.

Somewhere in a reference book I read about escape speed of a particle for earth.
Let a particle is projected from the Earth surface. Let its mass be m and speed of projection be u. Let mass of Earth be M and its radius be R.

According to law of conservation of energy,
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The problem is that,

If the Ist term becomes negative also but its magnitude is less than the IInd term, then also final Kinetic energy will be positive. And the particle will never doesn't give the proper answer. Friend isn't it so?
Please help me in understanding this.
Thank you all in advance.

 

[Simon Bridge]

The first term in what? You mean term I in the first relation?
That's kinetic energy - does it make physical sense to have a negative kinetic energy?

 

[hadronthunder]

The first term in what? You mean term I in the first relation?
That's kinetic energy - does it make physical sense to have a negative kinetic energy?

The first term is in the second relation. the complete 1/2(mu²) - (GMm)R

The complete term can be negative due to less value of 1/2 (mu²).

 

[Bandersnatch]

For escape velocity you're comparing the total energy at the surface of a massive body and at infinity from it.

This means that the second term $\frac{GMm}{R+h}$ will always be zero. So the first term can not be less in magnitude than the second one.If the first term becomes negative due to too low a value of initial velocity U, it simply means that the velocity U is too low to escape the gravity of the massive body.

 

[Nickie]

Hi there,

The concept of escape speed is an important one in the study of celestial bodies such as planets and moons. It refers to the minimum speed at which a particle must be launched from the surface of a planet in order to escape its gravitational pull and not fall back to the surface.

In the equations you have provided, the first term represents the potential energy of the particle at the Earth's surface, while the second term represents its kinetic energy. In order for the particle to escape, its final kinetic energy must be greater than or equal to its initial potential energy. This is because as the particle moves away from the Earth, its potential energy decreases while its kinetic energy increases.

If the first term becomes negative, it means that the potential energy of the particle at the Earth's surface is less than zero, which is not physically possible. This can happen if the particle is launched with a speed greater than the escape speed, causing it to have a negative potential energy.

In order to properly calculate the escape speed, you must use the correct values for the mass and radius of the Earth, and the mass of the particle. You can also use the equation v = √(2GM/R) where G is the gravitational constant, M is the mass of the Earth, and R is the radius of the Earth.

I hope this helps clarify the concept of escape speed for you. If you have any further questions, please don't hesitate to ask. Happy studying!