A \(2\,\mathrm{kg}\) mass is raised by \(5\,\mathrm{m}\). Find \(\Delta U_g\) using \(g=9.8\,\mathrm{m\,s^{-2}}\).

A \(3\,\mathrm{kg}\) object moves down \(4\,\mathrm{m}\). Find \(\Delta U_g\).

An object is lifted at constant speed. What happens to \(K\) and \(U_g\)?

If the zero of \(U_g\) is moved upward by \(2\,\mathrm{m}\), does \(\Delta U_g\) for the same motion change?

A \(0.50\,\mathrm{kg}\) ball falls \(8\,\mathrm{m}\) from rest. Find its impact speed, ignoring air resistance.

A \(4\,\mathrm{kg}\) mass gains \(196\,\mathrm{J}\) of gravitational potential energy. Find the height increase.

A \(1.5\,\mathrm{kg}\) object is thrown upward with speed \(10\,\mathrm{m\,s^{-1}}\). Find its maximum height above launch point, ignoring air resistance.

A \(2\,\mathrm{kg}\) object drops \(3\,\mathrm{m}\). Find the work done by gravity.

A cart starts from rest at height \(1.2\,\mathrm{m}\) above the bottom of a frictionless track. Find its speed at the bottom.

A \(0.20\,\mathrm{kg}\) ball is launched vertically upward at \(14\,\mathrm{m\,s^{-1}}\) from a balcony \(12\,\mathrm{m}\) above the ground. Find its speed just before it reaches the ground.

A block starts from rest a vertical height \(h\) above a horizontal table, slides on a frictionless track, then leaves the table horizontally from height \(H\). It lands a horizontal distance \(R\) from the table edge. Derive \(h\) in terms of \(R\) and \(H\), and state the assumptions needed for the result.

A bead is released from rest at height \(h\) above the bottom of a frictionless vertical circular loop of radius \(r\). Derive the regimes for \(h\): not enough energy to reach the top height, enough energy for the top height but not enough to stay in contact through the top, and enough to remain in contact.