Problem 4: A particle is fired with a velocity 16 km/s from the surface Earth. Find its velocity with which it moves in the interstellar space. Consider Earth’s escape velocity as 11.2 km/s and neglect friction.
Solution : We observe here that initial velocity of the particle is greater than Earth’s escape velocity. We can visualize this situation in terms of energy. The kinetic energy of the particle is used to (i) overcome the mechanical energy binding it to the gravitational influence of Earth and (ii) to move into interstellar space with a certain velocity.
Let “v”, “
v
e
v
e
” and “
v
i
v
i
” be velocity of projection, escape velocity and velocity in the interstellar space respectively. Then, applying law of conservation of energy :
1
2
m
v
2
−
G
M
m
R
=
1
2
m
v
i
2
1
2
m
v
2
−
G
M
m
R
=
1
2
m
v
i
2
⇒
1
2
m
v
2
=
G
M
m
R
+
1
2
m
v
i
2
⇒
1
2
m
v
2
=
G
M
m
R
+
1
2
m
v
i
2
Here, we have considered gravitational potential energy in the interstellar space as zero. Also, we know that kinetic energy corresponding to escape velocity is equal to the magnitude of gravitational potential energy of the particle on the surface. Hence,
⇒
1
2
m
v
2
=
1
2
m
v
e
2
+
1
2
m
v
i
2
⇒
1
2
m
v
2
=
1
2
m
v
e
2
+
1
2
m
v
i
2
⇒
v
i
2
=
v
2
−
v
e
2
⇒
v
i
2
=
v
2
−
v
e
2
The escape velocity for Earth is 11.2 km/s. Putting values in the equation, we have :
⇒
v
i
2
=
16
2
−
11.2
2
⇒
v
i
2
=
16
2
−
11.2
2
⇒
v
i
2
=
256

125.44
=
130.56
⇒
v
i
2
=
256

125.44
=
130.56
⇒
v
i
=
11.43
k
m
/
s
⇒
v
i
=
11.43
k
m
/
s