1.State Kepler's Three Laws of Planetary Motion.
Sol: 1. Law of Orbits: Elliptical orbits with Sun at one focus. 2. Law of Areas: Sweeps equal areas in equal intervals of time ($dA/dt = \text{const}$). 3. Law of Periods: $T^2 \propto a^3$.
2.Derive the expression for escape speed from the surface of Earth ($v_e = \sqrt{2gR}$).
Sol: Total energy at surface = $\frac{1}{2} m v_e^2 - \frac{GMm}{R} = 0 \implies v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR} \approx 11.2\text{ km/s}$.
3.Calculate acceleration due to gravity at a height $h=R$ above Earth's surface.
Sol: $g' = g \left(\frac{R}{R+h}\right)^2 = g \left(\frac{R}{2R}\right)^2 = \frac{g}{4} = \frac{9.8}{4} = 2.45\text{ m/s}^2$.
4.What is a geostationary satellite? State its height above Earth's surface and orbital period.
Sol: Satellite in equatorial orbit with period equal to Earth's rotation: $T = 24\text{ hours}$, height $h \approx 36,000\text{ km}$.
