Two cubes of the same mass but different sizes (smaller Cube A and bigger Cube B) are dropped from same height on to a tray of sand.Which cube will create a deeper impression and why?
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The smaller cube will create a deeper impression, since it has a smaller surface area impacting the sand but the same energy. To see this consider not dropping the cubes on sand but on an array of evenly spaced springs. It is easy to see that he spring will contract less if you can distribute the energy of the fall on more springs. The energy for both cubes dropped from a height $h$ is $E=mgh$ A springs potential Energy goes like $E=kx^2$ where $k$ is the spring constant. If you have $n$ springs under the surface they all will be displaced by $$E=nkx^2=mgh\Rightarrow x=\sqrt{\frac{mgh}{nk}}$$ from their equilibrium position As you see the more springs you have (i.e. more surface with evenly spaced springs) the smaller the 'impression'. PS: this is why spears und bullets and stuff like that are pointy at the top. All the energy they have will be focused on the small top, so they can exert much more pressure (force/area) and penetrate objects |
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both cubes come with the same speed as much downtime as the impact speed to not depend on mass, but the height. according to: $v^2 = 2gy$ (asumming that $V_0 = 0$ en $t =0$) Now, the depth left by the bins will depend on the energy that each one comes and this energy is proportional to the mass, ie, more mass, more energy of impact. according to: $$E = 1/2 m.V^2$$ Therefore, the "cube" of more mass is buried deeper than the other cube. |
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