Q.Six possible paths of a projectile launched near the Earth are drawn; in each the Earth is a circle with a dot marking its centre. The candidate paths are:

Concept: why the path is a focal conic

The gravitational force on the projectile always points towards the centre of the Earth and varies as $F = \dfrac{GM m}{r^2}$ (inverse square). It is a central, conservative force. Solving the motion under such a force gives a conic section with the force centre (the Earth's centre) at one focus — this is exactly Kepler's first law generalised to any projectile/satellite. So the single test is: is the drawn curve a conic that has the Earth's centre at a focus?

Testing each path

• (a) The loop is a closed ellipse, but the Earth's centre sits at the far end of it, not at a focus. A projectile cannot follow this. Not possible.
• (b) A closed ellipse with the Earth's centre at one focus. This is a bound (elliptical) orbit/trajectory. Possible.
• (c) Again a closed ellipse with the Earth's centre at a focus. Possible. …

Step 1: under gravity alone, any bound trajectory must be a conic section with Earth's centre at a FOCUS. Step 2: reject path (a) -- closed loop but Earth's centre at the extreme end, not a focus. Step 3: reject (d),(e),(f) -- inward spiral, non-conic curve, self-intersecting figure-eight ar …