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Rolling down

B1  3.00 Determine the horizontal range $x$ of the ball’s flight.

1 The axis is $R/\sqrt{2}$ away from the center 0.50
2 The moment of inertia of the ball
\[ I = \frac{2}{5} mR^2\]
0.20
3 The moment of inertia of the ball around the axis
\[ I = \frac{2}{5} mR^2 + \frac{mR^2}{2}\]
0.30
4 The law of conservation of energy:
\[ mgh = \frac{I \omega ^2}{2} \]
0.50
5 Constraint is written:
\[ v = \frac{\omega R}{\sqrt{2}}\]
0.40
6 The duration of flight:
\[ t = \sqrt{\frac{2H}{g}}\]
0.40
7 Horizontal motion equation:
\[ x = vt\]
0.20
8 \[x = \sqrt{\frac{20}{9}hH}\] 0.50