Conservation of Energy

Unit 3, Work, Energy, and Power, carries 18 to 23 percent of AP Physics 1. Energy is often the shortest route to a speed or a height, because it skips the time and the direction of motion entirely.

The energy forms you need

• Kinetic: K = ½mv²
• Gravitational potential: U_g = mgh, measured from a height you choose. Only changes in U_g matter.
• Spring potential: U_s = ½kx², with x measured from the spring's natural length.

When mechanical energy is conserved

If only gravity and spring forces do work, K + U stays constant:

K₁ + U₁ = K₂ + U₂

Normal forces on a fixed surface do no work, because they are perpendicular to the motion. A frictionless ramp or a swinging pendulum therefore conserves mechanical energy.

Adding friction

Kinetic friction turns mechanical energy into internal energy. The bookkeeping becomes:

K₁ + U₁ − f d = K₂ + U₂

Here f d is the energy converted by friction over a sliding distance d.

Energy bar charts

AP Physics 1 often asks for a bar chart before any calculation. Draw a bar for each form at the start and at the end, and one for the energy that leaves the system. The total height must match on both sides. The chart is where you show that you chose the system correctly.

Worked example

A 2 kg block starts from rest at the top of a ramp 5 m high. At the bottom it slides 4 m across a rough floor with a friction force of 5 N, then hits a spring with k = 800 N/m. Take g = 10 m/s². How far is the spring compressed?

1. Energy at the top: mgh = 2 × 10 × 5 = 100 J.
2. Energy lost on the rough floor: f d = 5 × 4 = 20 J, leaving 80 J.
3. The spring stores the rest: ½ × 800 × x² = 80, so x² = 0.2 and x ≈ 0.45 m.

A common mistake is to use the speed at the bottom of the ramp for the spring. The floor takes 20 J first.

The free response approach

• Name the system and say whether any external force does work on it.
• Write the energy equation symbolically, then substitute.
• Explain in words why a quantity is conserved or not. Saying there is no friction earns more than writing the equation alone.