Question

If a projectile has a velocity greater than the escape velocity, which trajectory will it follow?

Answer 1

answer : hyperbola

explanation : do you know about escape velocity of any planet. when we throw a object with escape velocity, object doesn't return to that planet, it means when we project an object with velocity equal to escape velocity ,the object will fly to infinity but after reaching there it would have no energy.

so it is the minimum velocity in fact to send an object to infinity.

however, if velocity of object is greater than escape velocity , then object will have some amount of kinetic when it reaches infinity.

so, E > 0

from derivation of Kepler's planetary mechanics,

we get, $e=\sqrt{1+\frac{2EL^2}{\mu(Gm_1m_2)^2}}$

where, e is eccentricity of conic section, E is total energy of body,

$\mu$ is effective mass of $m_1$ and $m_2$

e.g., $\mu=\frac{m_1m_2}{m_1+m_2}$.....(1) L is angular momentum and G is gravitational constant.

know come to mathematics,

according to conic section, if eccentricity greater than 1, trajectory will be hyperbola.

from above explanation we get if velocity of object greater than escape velocity then, E > 0

from equation (1),

e > 1 hence, trajectory will be hyperbola

Answer 2

The answer will be Hyberbola