Paris and London are connected by a straight subway tunnel (see Fig.). A train travels between the two cities powered only by the gravitational force of the earth. Calculate the maximum speed of the train and the time taken to travel from London to Paris. The distance between the two cities is 300 km and the radius of the earth is 6400 km. Neglect friction.
Text Solution
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Sol. Define x, h, r as in Fig. and assume the earth to be a stationary homogeneous sphere of radius R. Taking the surface of the earth as reference level, the gravitational potential energy of the train at x is
V = 
rdr = 
(r 2 – R 2 ),
Where m, M are the masses of the train and the earth respectively. Conservation of mechanical energy gives, as the train starts from rest at the earth's surface,
+ 
= 0,
or v 2 = 
,
where g = GM/R 2 is the acceleration of gravity at the earth's surface. As
r 2 = h 2 + (150 – x) 2 = (R 2 – 150 2 ) + (150 – x) 2 = R 2 – 300x + x 2 ,
v 2 = 
.
v is maximum when x = 150 km
v max = 
= 185.6 m/s.
The time from London to Paris is
T = 
= 
= 
= 42.3 min.
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