Only the component parallel to displacement contributes to work. A constant force gives a rectangular area. A linearly varying force adds a triangular contribution. Area below zero subtracts from work; a force perpendicular to displacement does no work.
The work-energy theorem connects net work to the change in kinetic energy. A brake does negative work because its force opposes displacement. The stopping distance is initial kinetic energy divided by brake force, so doubling speed quadruples stopping distance when mass and force are fixed.
Work is energy transferred. Power is the rate of that transfer. Both lifts raise the same mass through the same height and therefore do equal work against gravity. The lift with shorter duration needs greater average power. At constant speed the force is mg, giving P = mgv.
A spring stores energy as work is done against its restoring force. Because the force increases with deformation, spring energy is ½kx², not kx². Released on a frictionless platform, this energy alternates between elastic potential and kinetic forms. A raised platform adds constant gravitational potential mgh relative to the ground.
Gravity is conservative: its work depends only on the initial and final heights. The two routes have identical gravity work even when one climbs higher. A resisting force does negative work along every part of the journey, so its energy loss depends on total length. External work for slow transport supplies both the potential-energy increase and the dissipated energy.
Without friction, gravitational potential turns into kinetic energy and their sum stays constant. With friction, mechanical energy decreases but the energy balance includes dissipated thermal energy. A large enough resistance can prevent starting or stop an initially moving slider before the bottom.
Energy determines how speed changes while the bob climbs. Tension also must remain non-negative because a string cannot push. At the top the minimum taut speed is √(gr), which requires bottom speed √(5gr). A slower bob can turn back or lose string tension before reaching the top; the preview stops at that first limit.
An isolated collision conserves vector momentum. Restitution specifies how relative normal velocity changes; e = 1 is elastic and e = 0 removes relative normal motion. For smooth oblique contact only the normal components change, while tangential components remain unchanged. With no approaching normal motion the model applies no impulse.