Source: High school physics (Chinese)
Problem Sets:
Problem
A scientific experimental satellite moves in an elliptical orbit with the Earth's center at one focus. The altitude at perigee is $h_1 = 266$ km, and the altitude at apogee is $h_2 = 1826$ km. The satellite's speed at perigee is $v_1 = 8.13$ km/s. The Earth's radius is $R = 6.37 \times 10^3$ km. Air resistance is negligible.
v_2 = 6.58 km/s
Since the gravitational force on the satellite is a central force directed towards the Earth's center, the satellite's angular momentum about the Earth's center is conserved.
The distance from the Earth's center at perigee is $r_1 = R + h_1$. The distance from the Earth's center at apogee is $r_2 = R + h_2$.
By the conservation of angular momentum:
$$L_1 = L_2$$ $$m r_1 v_1 = m r_2 v_2$$Solving for the speed at apogee, $v_2$:
$$v_2 = \frac{r_1}{r_2} v_1 = \frac{R + h_1}{R + h_2} v_1$$Substituting the given values:
$$r_1 = 6370 \text{ km} + 266 \text{ km} = 6636 \text{ km}$$ $$r_2 = 6370 \text{ km} + 1826 \text{ km} = 8196 \text{ km}$$ $$v_2 = \frac{6636 \text{ km}}{8196 \text{ km}} (8.13 \text{ km/s})$$