Question

if an object is projected from the surface of the earth with speed k times the escape speed then interstellar speed ia​

Answer 1

When a body is projected from the surface of earth, If the speed of projection exceeds the escape velocity, the Object moves in interstellar space.

The speed of the object in the interstellar space is given by,

$v _\infty = \sqrt{v ^{2} -v _ e ^{2} }$

where,

$v _\infty , v, v _ e$

are respectively the interstellar velocity, velocity of projection, velocity of escape.

According to the question,

The object is projected with the speed k times the escape velocity.

$v = kv _ e$

So, The speed of the object in interstellar space is,

$v _\infty = \sqrt{v ^{2} -v _ e ^{2} } \\ \\ v _\infty = \sqrt{(kv _e)^{2} -v _ e ^{2} } \\ \\ v _\infty = \sqrt{(v _e)^{2} ( {k}^{2} - 1) } \\ \\ v _\infty = (\sqrt{ {k}^{2} - 1 } ) v_{e}$

Therefore, The speed of the object in interstellar space is

$v _\infty = (\sqrt{ {k}^{2} - 1 } ) v_{e} \:$