Answer to Question #150956 in Quantum Mechanics for Ron Yro Froid Alagos Gonzaga

Question

(a) What is the angular momentum of a particle of mass m that moves in a circle of radius r at the speed v? (b) An earth satellite follows an elliptical orbit in which its maximum altitude above the earth’s surface is 2000 km and its minimum altitude is 400 km. Use the result of (a) to find the ratio between the maximum and minimum speeds of the satellite.

Answer 1

1. Angular momentum in general is given by $\vec{L} = \vec{r}\wedge \vec{p}$ , where $\wedge$ denotes vector product. When particle moves in a circle, it's momentum is always orthogonal to it's position vector. Thus we have $L = p r = mvr$
2. Earth satellite conserves it's angular momentum and in addition when satellite attain it's lowest and highest points, momentum is orthogonal to the position vector. Thus we have (we need to add Earth's radius to the altitude) :

$mv_{low} (R+h_{min}) = mv_{high}(R+h_{max})$

And we find $\frac{v_{low}}{v_{high}} = \frac{R+h_{max}}{R+h_{min}} = \frac{8400}{6800}\approx 1.24$ .