Katamari damacy: make a giant ball of crap before time is up!

Calculus Level 5

Katamari Damacy is a popular Japanese video game where a character rolls a ball around a world that is filled with random things. As the ball rolls it picks up objects in its path and engulfs them, resulting in a growth process.

To a first approximation, any object of volume $V$ that is picked up is instantly and uniformly distributed to the volume of the growing ball of crap. Also, the ball rolls with constant angular velocity.

Suppose that in a simple world, the character rolls the ball in a straight line that has random crap distributed with a line density of $\lambda_0$ kg/m. If the original radius doubles in time $T_2$ , and it increases eight-fold in time $T_8$ , what is the value of $T_8/T_2?$

The answer is 21.

2 solutions

Ayush Verma
Oct 22, 2014

$\cfrac { dV }{ dr } =4\pi { r }^{ 2 },\\ \\ { \lambda }_{ 0 }=\cfrac { dm }{ dx } =\rho \cfrac { dV }{ dx } \\ \\ \Rightarrow \cfrac { dV }{ dx } =\cfrac { { \lambda }_{ 0 } }{ \rho } \\ \\ \cfrac { dx }{ dt } =r\omega =\cfrac { dx }{ dV } \cfrac { dV }{ dt } =\cfrac { \rho }{ { \lambda }_{ 0 } } 4\pi { r }^{ 2 }\cfrac { dr }{ dt } \\ \\ \Rightarrow 2r\cfrac { dr }{ dt } =\cfrac { { 2\lambda }_{ 0 }\omega }{ 4\pi \rho } =k\quad (constant)\\ \\ \Rightarrow k{ T }_{ 2 }={ \left[ { r }^{ 2 } \right] }_{ r }^{ 2r }=3{ r }^{ 2 }\quad \& \quad k{ T }_{ 8 }={ \left[ { r }^{ 2 } \right] }_{ r }^{ 8r }=63{ r }^{ 2 }\\ \\ \Rightarrow \cfrac { { T }_{ 8 } }{ { T }_{ 2 } } =\cfrac { 63 }{ 3 } =21$

Vishal Sharma
Oct 14, 2014

Let in small time dt change in radius be dr.Change in volume is dv. So dv=4πr2 dr.Let angular speed be w and let density of material be D. Then (wrdt Line density)/dv=D.Put dv in form of dr and integrate on both sides.In integration limits assume initial radius to be r1 and final be r. T8/T2=126/6=21

Katamari damacy: make a giant ball of crap before time is up!

The answer is 21.

2 solutions

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