a particle of mass m is projected in a transverse direction from a height R ( Radius of earth ) above surface of earth find ghe minimum initial speed so that it just grazes the surface of earth
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Answer:
Suppose velocity of projection at point A is v
A
and at point B, the velocity of the particle is v
B
.
Then applying Newton's second law at points A and B, we get
ρ
A
mv
A
2
=
(R+h)
2
GM
e
m
and
ρ
B
mv
2
−
R
2
GM
e
m
where ρ
A
and ρ
B
are radii of curvature of the orbit at points A and B of the ellipse, but ρ
A
=ρ
B
=ρ(say).
Now applying conservation of energy at points A and B,
R+h
−GM
e
m
+
2
1
mv
A
2
=−
R
GM
e
m
+
2
1
mv
B
2
⇒GM
e
m(
R
1
−
(R+h)
1
)=
2
1
(mv
B
2
−mv
A
2
)
=(
2
1
ρGM
e
m(
R
2
1
−
(R+h)
2
1
))
or ρ=
2R+h
2R(R+h)
=
R+r
2Rr
Thus we get
V
A
2
=
(R+h)
2
ρGM
e
=2GM
e
r(r+R)
R
where r=distance of point of projection from earth's centre =R+h.
Explanation:
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