Suppose that a projectile is fired straight upward from the surface of the earth

Question

Suppose that a projectile is fired straight upward from the surface of the earth with initial velocity Vo and assume its height y(t) above the surface satisfies the following IVP
{
y''=-GM/(y+R)^2 - Be^-y
y(0)=0
y'(0)=Vo
}
Please compute the total amount of time for the projectile to reach at the highest point from launch and velocity at the instance when the projectile re-touches the earth surface. Compute the velocity at the instant when the projectile retouches the ground
(G,M,R,Vo,B are constants which compose the answers)

Explanation / Answer

y'' = a = dv /dt
do separation of variables and integrate to find v (t) = y' = dy/dt
You can do separation of variables again , and to solve for y(t)
before you have y'' = dv/dt = a
now lets substitute dv/dt = v( dv/dy)
Plug that into the equation and NOW do separation of variables

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