Newtonian mechanics

Conservation of mechanical energy

When only conservative forces do work, a system's kinetic plus potential energy remains constant: K+U=const.

A falling ball speeds up as gravitational potential energy decreases. Neglecting air resistance, that lost potential energy becomes kinetic energy, leaving total mechanical energy conserved.

E_{\rm m}=K+U=\text{constant}

Definition: Mechanical energy

Mechanical energy $E_m=K+U$ is the sum of a system's kinetic and potential energies. If nonconservative forces (friction, drag) do no work, $K_i+U_i=K_f+U_f$ . When they do work, $\Delta(K+U)=W_{\rm nc}$ ; mechanical energy can become heat or other forms, while total energy is still conserved.

Watch a ball exchange height and speed; add friction to see mechanical energy transferred to heat.

Applying conservation

Choose two states, write $K_i+U_i=K_f+U_f$ , and use one consistent potential-energy reference. For an object released from rest and falling height $h$ without drag, $mgh=\frac12mv^2$ , so $v=\sqrt{2gh}$ . Mass cancels, so the model predicts the same fall speed for any mass.

Example: Speed after a fall

An object is released from rest 5 m above the ground. Neglect drag and take $g=10\,m/s^2$ . Find its speed just before impact.

Solution

$v=\sqrt{2gh}=\sqrt{2\times10\times5}=10\,m/s$ .

To use mechanical-energy conservation, select two states, keep one potential-energy reference, and check whether drag or friction does significant work. A body sliding on a smooth track converts potential energy into kinetic energy; on a rough track, some mechanical energy becomes thermal energy through friction. If initial mechanical energy is 50 J and friction transfers 8 J away, final mechanical energy is 42 J, while total energy of object, track, and surroundings remains conserved.

Example: A frictionless slide

A child starts from rest at height 5 m and slides without friction. Take g=10 m/s². Find speed at the bottom.

Solution

mgh=½mv², so v=√(2gh)=√100=10 m/s.

A useful first step is to choose the lowest point as the zero of potential energy, then write initial and final energies using that same reference. If the initial speed is not zero, keep its kinetic-energy term too. Conservation simplifies calculations but does not replace checking for nonconservative forces.

Quick check

For a dropped object with no drag, what happens to kinetic energy as potential energy falls?

A sliding object slows due to friction. Into what form is much of the lost mechanical energy transferred?

References

Young, Freedman (2019). University Physics