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Imagine a carnival game, the ones where you have to shoot down items with a toy gun, my question is where to shoot the item to maximize the rotational moment of the item. My studies lead me to believe it is at the highest point of the item where the perpendicular distance is the greatest, which will make the item topple backwards, but what if you shoot it as close to the base as possible? Will it topple forward more easily?

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It could topple forward just as easily as it would backwards, but only in very ideal situations. Practically there will be problems. Assuming an object of uniform width --->

Our projectile has a definite size, hence it is easy to just aim it slightly above the top of the object, and it hits the top nearly perpendicularly. If you aim it towards the bottom, the projectile will hit the surface on which the object is kept.

Also, consider what friction would do. Although it will not have much of an impact when the impulsive interactions (projectile hitting object) dominate, it might be the difference maker if the projectile is rather slow. When hitting at the top, the friction on the base prevents the body from sliding off and while doing so it provides extra angular velocity to the body in the same direction as is provided by the impulse from the projectile. When hitting at the bottom, the friction gives you an opposite effect -- it reduces the angular velocity given by the projectile.

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