Physics - Gravitation Question with Solution | TestHub

Option A: For an elliptical orbit, the total mechanical energy is given by $E=-\frac{GMm}{2a}$, where $a$ is the semi-major axis. This is a standard result from orbital mechanics. This is **correct**.

Option B: The magnitude of the angular momentum for an elliptical orbit is given by $L=m\sqrt{GMa(1-e^2)}$. This is a standard formula derived from conservation of angular momentum and Kepler's laws. This is **correct**.

Option C: The gravitational force is a central force, so the angular momentum of the satellite about the center of the planet is conserved. At perigee and apogee, the velocity vector is perpendicular to the position vector, so the magnitude of angular momentum is $L=mvR$. Thus, $m{v}_P{R}_P=m{v}_A{R}_A$, which implies $v_P{R}_P=v_A{R}_A$. This is **correct**.

Option D: For an elliptical orbit, the total mechanical energy $E<0$. If the satellite's speed is increased such that its total mechanical energy becomes exactly zero ($E=0$), its trajectory changes from elliptical to parabolic. If $E>0$, the trajectory would be hyperbolic. This is a fundamental concept related to escape velocity. This is **correct**.