A projectile is fired from the earth with the aim at getting into space. The velocity (v) of the projectile is described by the following ODE: (dv)/(dr)=-(gR^(2))/(vr^(2)) Where r is the distance from the centre of the earth to the projectile, g is the earth's gravity and R is the radius of the earth. The initial velocity is denoted v_(0) and occu

Question

A projectile is fired from the earth with the aim at getting into space. The velocity

(v)

of the projectile is described by the following ODE:

(dv)/(dr)=-(gR^(2))/(vr^(2))

Where

r

is the distance from the centre of the earth to the projectile,

g

is the earth's gravity and

R

is the radius of the earth. The initial velocity is denoted

v_(0)

and occurs on the surface of the earth. a) Solve the above ODE and explicitly express velocity

(v)

as a function of

r

b) Show with justification that the minimum velocity required to ensure that the object does not return to earth is given by:

v=\sqrt(2gR)

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