Problematic Balloon

Help me with this one.

A balloon moves up vertically such that if a stone is projected with a horizontal velocity $u$ relative to balloon, the stone always hits the ground at a fixed point at a distance $\frac{2u^2}{g}$ horizontally away from it. Find the height of the balloon as a function of time.

The answer is

$\frac{2u^2}{g}$ ( $1-e^{-gt/2u}$ )

#Mechanics

Easy Math Editor

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Comments

Well this is not very difficult.
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 of $u$ 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 $\displaystyle \frac{2u^2}{g}$ . Hence if the balloon takes a time $t_0$ to reach the ground then we can easily write $\displaystyle u t_0 = \frac{2u^2}{g} \Rightarrow t_0 = \frac{2u}{g}$ . Also using the second equation of motion in the vertical direction, we obtain $\displaystyle h = -v t_0 + \frac{1}{2} gt_0^2$ . Substitute the value of $t_0$ to obtain an equation of the form

$\displaystyle \Rightarrow v + \frac{gh}{2u} = u$
$\displaystyle \Rightarrow \frac{\text{d}h}{\text{d}t} + \frac{gh}{2u} = u$
This is a standard differential equation with the solution of the form $\displaystyle h.\text{exp}(\frac{gt}{2u}) = ut + C$ , where $C$ is the constant of integration. Rearrange to obtain the final answer as $h = (ut + C) \text{exp}(\frac{-gt}{2u})$

(Note: $\text{exp}(x) = e^x$ )
Hope this helps.

Sudeep Salgia - 6 years, 11 months ago

I got a different differential equation than yours. It is

$h$ $=$ $\frac{2u^2}{g}$ + $c$ $e^{(-gt/2u)}$

I am just having problem in finding the constraint. I am confused whether to take

when $t$ $=$ $t_{0}$ , $h$ $=$ $0$

or, when $t$ $=$ $0$ , $h$ $=$ $0$

Akhilesh Prasad - 6 years, 11 months ago

from which source did u get the problem

wrik ray - 5 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$