Conservation of Mechanical Energy

Core idea

Mechanical energy changes form while the total is tracked

Mechanical energy is the sum of the kinetic and potential energies included in the chosen system. In a closed model with only conservative internal interactions, energy moves between forms while the total mechanical energy stays constant.

E_mech = K + U_g + U_s
K_i + U_i = K_f + U_f

Friction and an applied push can transfer energy across the system boundary. Mechanical energy can then decrease or increase even though total energy is still conserved in a larger accounting system. Start by naming the system, the initial and final states, and the reference level.

Guided lesson path

Follow energy from the starting state to the final state

1. Identify the stores. List kinetic energy, gravitational potential energy, and elastic potential energy that belong to the chosen system.
2. Write the conservation equation. For a frictionless system, use K_i + U_i = K_f + U_f and state which potential-energy terms are present.
3. Recognize the condition. Mechanical energy is conserved only when the chosen model has no non-conservative energy transfer across its boundary.
4. Choose the boundary, states, and reference. Decide which objects are included, record initial and final positions and speeds, and set U_g = 0 at a convenient level.
5. Predict the frictionless bowl. A skater released high on one side converts gravitational potential energy into kinetic energy while descending.
6. Watch U_g decrease. As height falls, gravitational potential energy decreases by the same amount that kinetic energy increases.
7. Watch K increase. Speed is greatest near the bottom because the skater has converted the largest available amount of U_g into K.
8. Test the total. Add K and U_g at several positions. The total should remain approximately constant in the frictionless run.
9. Interpret the bottom state. At the lowest point, U_g is smallest relative to the reference and K is largest, but the mechanical-energy total is unchanged.
10. Follow energy to the opposite side. The skater slows as K changes back into U_g. Equal heights on opposite sides correspond to equal speeds in the ideal model.
11. Explain the normal force. The bowl’s normal force changes the direction of velocity but does no work when it remains perpendicular to the path, so it does not change the mechanical-energy total.
12. Predict a vertical drop. For a 2 m drop from rest, gravitational work changes kinetic energy by mgh = (m)(9.8)(2). The mass cancels when solving the final speed.
13. Measure ΔU_g. A downward 2 m change gives ΔU_g = −mgh. The negative potential-energy change matches positive gravitational work.
14. Measure ΔK. In the frictionless drop, ΔK = +mgh, so ΔK + ΔU_g = 0.
15. Calculate the final speed. From rest after a 2 m drop, K_f = mgh and v_f = √(2gh) ≈ 6.26 m/s, independent of mass.
16. Use height–speed reasoning. A larger vertical drop provides more available gravitational potential energy and therefore a larger final speed in the same ideal model.
17. Test mass independence. Changing mass scales K and U_g by the same factor, so the maximum height and speed relationships can remain unchanged.
18. Analyze a projectile state. At the apex of a launch, vertical velocity can be zero while horizontal kinetic energy remains. Include both K and U_g in the energy equation.
19. Verify the projectile accounting. Compare kinetic energy at launch and at maximum height with the corresponding gravitational potential-energy increase. The mechanical-energy total should agree.
20. Finish with the decision test. If the total mechanical energy changes, identify the external work, friction, or other non-conservative transfer responsible before declaring energy “lost.”

Worked examples

Write the initial and final energy states

For a 2 kg object released from rest and dropping 2 m:

K_i + U_gi = K_f + U_gf
0 + mgh = ½mv_f² + 0
v_f = √(2gh) = √(2·9.8·2) = 6.26 m/s

The energy changes show the same result:

ΔU_g = (2 kg)(9.8 m/s²)(−2 m) = −39.2 J
ΔK = +39.2 J;   ΔE_mech = ΔK + ΔU_g = 0 J

For an object launched from ground level with horizontal and vertical velocity components, use total kinetic energy at launch and at the apex:

K = ½m(v_x² + v_y²);   at the apex, v_y = 0 but v_x can remain nonzero.

System boundaries

“Conserved” depends on what the system includes

If the skater and Earth are included, gravitational potential energy is an internal store and gravity transfers energy between U_g and K. If Earth is outside the system, gravitational work crosses the boundary and must appear as an energy transfer.

With friction, the skater–Earth system can still conserve total energy while mechanical energy decreases. The missing mechanical energy has become thermal energy at the contact. A changing E_mech is evidence of a transfer or transformation, not energy disappearing.

Reference levels

Changing U_g = 0 does not change motion

Choose a zero level that makes the states easy to describe. Every U_g value shifts when the reference moves, but differences between the same two heights do not:

ΔU_g = mg(y_f − y_i)

Use the same reference in the initial and final terms of one equation. Mixing reference levels can create an artificial energy imbalance even when the physical model is correct.

Friction and energy

Mechanical energy can decrease while total energy is conserved

Friction does negative work on the moving object. In a mechanical-energy-only account, that lowers E_mech. In a larger energy account that includes the surfaces, the same amount appears as thermal energy generated at the contact.

This is why a skater with friction does not return to the same height on the opposite side. Some of the initial mechanical energy has been transferred into thermal energy, so the next turning point is lower.

For teachers

Make the energy total a measured claim

Start by asking students to list the energy stores before they run the frictionless bowl. Have them record K, U_g, and E_mech at the release point, bottom, and opposite side instead of treating “conservation” as a slogan.

Next, add friction and ask where the mechanical-energy decrease went. Finish with the 2 m drop and projectile apex trials so students see mass cancellation, horizontal kinetic energy, and the importance of the system boundary.

Continue with Conservative Forces and GPE for path independence and reference levels, Work and Kinetic Energy for the work–energy theorem, and the Conservation of Mechanical Energy simulation for the full measurement sequence.