Many mechanics problems can be solved without directly using Newton’s second law. If the forces are conservative,
the total mechanical energy remains constant. This is especially useful for curved ramps, projectiles, and pendulums.
1. Kinetic energy
Kinetic energy is the energy of motion:
\[
K=\frac12mv^2.
\]
It depends on mass and the square of speed. Doubling speed makes kinetic energy four times larger.
2. Gravitational potential energy
Near Earth’s surface, gravitational potential energy is
\[
U_g=mgh.
\]
The height \(h\) is measured relative to a chosen zero level. Only height differences matter in most energy problems.
3. Conservation of mechanical energy
If only gravity and other conservative forces act, then
\[
K_i+U_i=K_f+U_f.
\]
Substituting the formulas for \(K\) and \(U_g\),
\[
\frac12mv_i^2+mgh_i
=
\frac12mv_f^2+mgh_f.
\]
The mass cancels for ideal gravitational motion, so the speed depends on height change, not on mass.
4. Including non-conservative work
If friction, drag, a motor, or another non-conservative effect acts, use the more general balance:
\[
K_i+U_i+W_{\mathrm{nc}}=K_f+U_f.
\]
Here \(W_{\mathrm{nc}}\) is signed. Friction and drag are usually negative. External work from a motor or push can be positive.
\[
W_{\mathrm{nc}}=E_f-E_i.
\]
5. Useful rearranged equations
| Unknown |
Formula |
Use |
| Final speed |
\(v_f=\sqrt{v_i^2+2g(h_i-h_f)+\frac{2W_{\mathrm{nc}}}{m}}\) |
Find speed after falling, sliding down, or swinging lower. |
| Initial speed |
\(v_i=\sqrt{v_f^2+2g(h_f-h_i)-\frac{2W_{\mathrm{nc}}}{m}}\) |
Find launch speed or starting speed. |
| Final height |
\(h_f=h_i+\frac{v_i^2-v_f^2}{2g}+\frac{W_{\mathrm{nc}}}{mg}\) |
Find maximum height or target height. |
| Initial height |
\(h_i=h_f+\frac{v_f^2-v_i^2}{2g}-\frac{W_{\mathrm{nc}}}{mg}\) |
Find required starting height. |
| Non-conservative work |
\(W_{\mathrm{nc}}=K_f+U_f-K_i-U_i\) |
Find work lost to friction/drag or added by an external agent. |
6. Ramp applications
For a frictionless ramp, the exact shape of the ramp does not matter. Only the vertical height change matters:
\[
\frac12mv_i^2+mgh_i=\frac12mv_f^2+mgh_f.
\]
If an object starts from rest at height \(h_i\) and reaches the bottom at \(h_f=0\), then
\[
v_f=\sqrt{2gh_i}.
\]
7. Projectile applications
For projectile motion without air resistance, energy gives the speed at any height:
\[
\frac12mv_i^2+mgh_i=\frac12mv_f^2+mgh_f.
\]
Energy does not directly give the direction of velocity. It gives the speed magnitude. Projectile direction still depends on components.
8. Pendulum applications
A pendulum transforms gravitational potential energy into kinetic energy as it swings downward:
\[
mgh=\frac12mv^2.
\]
So the speed at the bottom is
\[
v=\sqrt{2gh},
\]
where \(h\) is the vertical drop from the release point to the bottom.
9. Worked example: curved ramp
An object starts from rest at a height of \(25\ \mathrm{m}\) and slides down a frictionless curved ramp to the bottom.
The initial speed is \(v_i=0\), the final height is \(h_f=0\), and \(W_{\mathrm{nc}}=0\).
\[
\frac12mv_i^2+mgh_i=\frac12mv_f^2+mgh_f.
\]
Since \(v_i=0\) and \(h_f=0\),
\[
mgh_i=\frac12mv_f^2.
\]
Cancel \(m\):
\[
gh_i=\frac12v_f^2.
\]
Solve for \(v_f\):
\[
v_f=\sqrt{2gh_i}.
\]
Substitute \(g=9.81\ \mathrm{m/s^2}\) and \(h_i=25\ \mathrm{m}\):
\[
v_f=\sqrt{2(9.81)(25)}\approx22.1\ \mathrm{m/s}.
\]
Therefore,
\[
\boxed{v_f\approx22.1\ \mathrm{m/s}}.
\]
Formula summary
| Concept |
Formula |
Meaning |
| Kinetic energy |
\(K=\frac12mv^2\) |
Energy due to motion. |
| Gravitational potential energy |
\(U_g=mgh\) |
Energy due to height in a gravitational field. |
| Mechanical energy |
\(E_{\mathrm{mech}}=K+U_g\) |
Total kinetic plus gravitational potential energy. |
| Conservative energy balance |
\(K_i+U_i=K_f+U_f\) |
Use when friction, drag, and external work are negligible. |
| General energy balance |
\(K_i+U_i+W_{\mathrm{nc}}=K_f+U_f\) |
Use when non-conservative forces act. |
Energy methods are powerful because the path shape often does not matter. For gravity-only motion, the speed depends on height change, not the detailed path.
