Dot Point from syabus- need help (1 Viewer)

[OMGRX7]

yo, i was searching up the ressources area, and like i couldnt find what i needed, and also several text books, so now im trying to get someone to explain it to me.

* explain the concept of escape velocity
in terms of the:
– gravitational constant
– mass and radius of the planet



yeah, i need someone to explain that to me. thanks

 

[Dark Phoenix]

escape velocity is the velocity required by a projectile to rise up vertically and just escape the gravitational field of a planet.

By equating the kinetic energy (1/2 mv^2) and gravitational energy of a projectile (which is (Gmm/r)), it can be shown that the escape velocity of a planet depends only upon the Universal Gravitational constant, the mass and the radius of the planet

when equating the 2 formulas, the result should be √(2Gm/r)

 

[Forbidden.]

The escape velocity

-Is inversely proportional to the square root of the radius of the planet.
-Is proportional to the square root of the mass of the planet

There's a fair bit to talk about the formula v² = √Gm/r

Now theoretical physicists believe if the universal gravitational constant G would change, so would the escape velocity, proportionally.
But all constants really don't change.




How to derive escape velocity:

E_k + E_p = 0

From Preliminary Physics you know when Kinetic Energy decreases the Gravitational Potential Energy increases and vice versa.
Know your formulas, well not just the E_p = mgh one, but the E_p = -Gm₁m₂/r

1/2mv² + (-Gm₁m₂/r) = 0

Place -Gm₁m₂/r on RHS by adding it on both sides, cancelling the LHS one.

1/2mv² = Gm₁m₂/r

Cancel out the mass from LHS and either (not both) mass from RHS. Subscripts 1 and 2 not needed on RHS.

1/2v² = Gm/r

1/2 * 2 = 2/2 = 1, so multiply both sides by 2

v² = 2Gm/r

Square root both sides.

v = √2Gm/r