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the potential energy of a golf ball in a hole is negative with respect to the ground. Under what circumstances(if any) is the ball's kinetic energy negative? Its rest energy?

The kinetic energy of the ball is zero when it isnt moving, whether it is ion the hole or not. Kinetic energy cannot be negative, since V^2 is positive or zero.

Since rest energy = m c^2, and c^2 is a huge number, and the mass m is non-zero, rest energy cannot be negative either.

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