Find the speed of a satellite at a distance R from Earth

[Helly123]

So, the total mechanical energy
= Ek + Ep = 1/2mv^2 + (-GmM/r) = gRm/4 - GmM/2R = -mgR/2. Is it?

 

[ehild]

So, the total mechanical energy
= Ek + Ep = 1/2mv^2 + (-GmM/r) = gRm/4 - GmM/2R = -mgR/2. Is it?

No. Do not mix g and G. You know that GM/R²=g, so GM=gR².

 

[Helly123]

No. Do not mix g and G. You know that GM/R²=g, so GM=gR².

Yes

 

[ehild]

Yes

So what result did you get?

 

[PeterO]

So, r = 2R
r is distance from satellite to Earth's center
Btw, mechanical energy is Ek. The answer is -mGR/4
How can Ek be negative?

NO. The Mechanical Energy is the sum of the Gravitational Potential Energy plus the Kinetic Energy. The Kinetic Energy may be positive, but the Mechanical Energy value can still be negative, provided the Potential Energy is negative, and larger in size than the Kinetic Energy.

 

[Helly123]

Ok. My answer is like Ep + Ek
While the formula i know for Ep is -GMm/r
r = 2R
Ek as usual is 1/2mv^2
V = $\sqrt{ gmR/2}$

Ep + Ek
-GMm/2R + mgR/4 = -mgR/2 + mgR/4 = -mgR/4
Is that it?

 

[Helly123]

NO. The Mechanical Energy is the sum of the Gravitational Potential Energy plus the Kinetic Energy. The Kinetic Energy may be positive, but the Mechanical Energy value can still be negative, provided the Potential Energy is negative, and larger in size than the Kinetic Energy.

NO. The Mechanical Energy is the sum of the Gravitational Potential Energy plus the Kinetic Energy. The Kinetic Energy may be positive, but the Mechanical Energy value can still be negative, provided the Potential Energy is negative, and larger in size than the Kinetic Energy.

I see

 

[ehild]

Ok. My answer is like Ep + Ek
While the formula i know for Ep is -GMm/r
r = 2R
Ek as usual is 1/2mv^2
V = $\sqrt{ gmR/2}$

Ep + Ek
-GMm/2R + mgR/4 = -mgR/2 + mgR/4 = -mgR/4
Is that it?

Yes.

 

[Helly123]

Yes.

Ok

 

[Dr Dr news]

The potential energy is negative and twice the magnitude of the kinetic energy (which is positive), thus the total energy is negative. So work is required to escape the gravitational well of the Earth no matter where you are.