The discussion focuses on calculating the speed of a satellite at a distance R from the Earth's surface, utilizing gravitational force equations. Key equations include the orbital velocity formula \( v = \sqrt{gR} \) and escape velocity \( v = \sqrt{2gR} \). Participants clarify that for a circular orbit, the centripetal force must equal gravitational force, leading to the conclusion that the correct orbital speed is \( v = \sqrt{GM/R} \). The total mechanical energy of the satellite is also discussed, emphasizing that it can be negative due to the gravitational potential energy being defined as zero at infinity.
PREREQUISITESStudents studying physics, particularly those focusing on mechanics and gravitational forces, as well as educators looking for examples of satellite motion calculations.
No. Do not mix g and G. You know that GM/R2=g, so GM=gR2.Helly123 said:So, the total mechanical energy
= Ek + Ep = 1/2mv^2 + (-GmM/r) = gRm/4 - GmM/2R = -mgR/2. Is it?
Yesehild said:No. Do not mix g and G. You know that GM/R2=g, so GM=gR2.
So what result did you get?Helly123 said:Yes
Helly123 said: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?
PeterO said: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 seePeterO said: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.
Yes.Helly123 said: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?
Okehild said:Yes.