(a) A rocket of (variable) mass m is propelled by steadily ejecting part of its mass at velocity u (constant with respect to the rocket). Neglecting gravity, the differential equation of the rocket is m ( d v / d m ) = − u as long as v ≪ c , c = speed of light. Find v as a function of m if m = m 0 when v = 0. (b) In the relativistic region ( v / c not negligible), the rocket equation is m d v d m = − u 1 − v 2 c 2 . Solve this differential equation to find v as a function of m. Show that v / c = ( 1 − x ) / ( 1 + x ) , where x = m / m 0 2 u / c .
(a) A rocket of (variable) mass m is propelled by steadily ejecting part of its mass at velocity u (constant with respect to the rocket). Neglecting gravity, the differential equation of the rocket is m ( d v / d m ) = − u as long as v ≪ c , c = speed of light. Find v as a function of m if m = m 0 when v = 0. (b) In the relativistic region ( v / c not negligible), the rocket equation is m d v d m = − u 1 − v 2 c 2 . Solve this differential equation to find v as a function of m. Show that v / c = ( 1 − x ) / ( 1 + x ) , where x = m / m 0 2 u / c .
(a) A rocket of (variable) mass
m
is propelled by steadily ejecting part of its mass at velocity
u
(constant with respect to the rocket). Neglecting gravity, the differential equation of the rocket is
m
(
d
v
/
d
m
)
=
−
u
as long as
v
≪
c
,
c
=
speed of light. Find
v
as a function of
m
if
m
=
m
0
when
v
=
0.
(b) In the relativistic region (
v
/
c
not negligible), the rocket equation is
m
d
v
d
m
=
−
u
1
−
v
2
c
2
.
Solve this differential equation to find
v
as a function of m. Show that
v
/
c
=
(
1
−
x
)
/
(
1
+
x
)
,
where
x
=
m
/
m
0
2
u
/
c
.
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