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Let the velocity of the balloon when it is at a height h be v. Hence when the stone is thrown from the balloon with a velocity u of in the horizontal direction, relative to the balloon, the actual velocity of the stone, with respect to the ground will be u in the horizontal direction and v in the vertical. Now, they say that irrespective of the height of the balloon above the ground the range of the stone is 2u^2/g. Hence if the balloon takes a time t0 to reach the ground then we can easily write .
ut0=2u^2/g
t0=2u^2/ug=2u/g
Also using the second equation of motion in the vertical direction, we obtain . Substitute the value of to obtain an equation of the form
h=-vt0+1/2gt^2
substituing the value of t0
h=-v(2u/g)+1/2gt^2
v+gh/2u=u
dh/dt+gh/2u=u
This is standard differential equation with soltuin h(exp)(gt/2u)=ut+C
where C is constant or integration
h=ut+Cexp)(-gt/2u)
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