# How high will the Ball Jump? [closed]

1. A small ball of density $\rho$ is immersed in a liquid of density $\sigma$, ($\sigma>\rho$) to the depth $h$ and released. How can we find the height above surface of the liquid up to which the ball will jump?

2. And what if we change the question so that the ball is dropped from a height of $h$ how deep in the liquid the ball will go? Will the this will be solved with the same method as the first question?

3. Will the ball continue what is called a simple harmonic motion if surface tension and Drag is ignored?

My work

1.Net upward force acting on the body; $F$=$\sigma vg-\rho vg$=$ma$

$a=g(\frac{\sigma}{\rho}-1)$

velocity after height $h$; $v=\sqrt{2g(\frac{\sigma}{\rho}-1)h}$

conserving mechanical energy at max height $h'$ and at surface,

$mgh'$=$\frac{1}{2}m(2g(\frac{\sigma}{\rho}-1)h)$

$\fbox{$h'=(\frac{\sigma}{\rho}-1)h$}$

1. velocity just before touching the liquid= $\sqrt{2gh}$

negative acceleration= $g(\frac{\sigma}{\rho}-1)$ from part 1.

using, $v^2=u^2-2g(\frac{\sigma}{\rho}-1)h'$

$\fbox{$h'=\frac{h}{(\frac{\sigma}{\rho}-1)}$}$

• Hi Dheeraj Kumar. Welcome to Phys.SE. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. Commented Dec 12, 2014 at 12:35
• BTW @Qmechanic can you answer Part 3 please Commented Dec 12, 2014 at 12:40
• It won't be simple harmonic, because one of the restoring forces (buoyancy) changes at the water-air interface, while gravity remains constant. It will be some sort of harmonic behavior so long as friction, surface tension, etc. are ignored. Commented Dec 12, 2014 at 12:53
• @CarlWitthoft I tried and found that it will do a SHM for $\sigma$=2, $\rho$=1, and initial depth=1 unit, you can check it by putting different sets of values in part 1 and 2 Commented Dec 12, 2014 at 13:03
• What's your density for air? Are you using zero, aka vacuum? Commented Dec 12, 2014 at 13:08

First you must find the speed when the ball comes out of water, the same way you calculate the speed of free fall of an item in the air. $$m\vec{a}=\Sigma{\vec{F_i}}$$ where the forces are : friction, weight, Archimedes force.