AP Physics C Mechanics — Potential Energy & Energy Conservation (Unit 3)

Kinetic Energy

What kinetic energy is

Kinetic energy is the energy an object has because it is moving. In mechanics, it is a scalar quantity (no direction) that depends on the object’s mass and speed. The standard expression is

K = \frac{1}{2}mv^2

where K is kinetic energy, m is mass, and v is speed.

A key idea: kinetic energy is not “something extra” added onto motion—it is a way to account for motion in an energy bookkeeping system. When forces do work on an object, they change its kinetic energy. That connection is what makes energy methods so powerful on AP Physics C problems.

Why kinetic energy matters

Kinetic energy matters because it links dynamics (forces and acceleration) to kinematics (speeds and positions) without requiring you to track time. Many AP problems involve complicated paths, changing forces, or multi-stage motion. Instead of solving a differential equation in time, you can often compare energies between two positions.

Kinetic energy is also the “currency” that potential energy can convert into. When you let go of a compressed spring or drop an object, you typically track how potential energy becomes kinetic energy.

How kinetic energy works: the Work–Kinetic Energy Theorem

The central relationship connecting force and kinetic energy is the work–kinetic energy theorem:

W_{\text{net}} = \Delta K = K_f - K_i

Here W_{\text{net}} is the net work done by all forces on the object. This theorem is foundational because it tells you that you can find speed changes by computing work—often easier than finding acceleration as a function of time.

For a constant force parallel to displacement, work is

W = Fd

More generally, if the force varies with position or direction, work is defined via the line integral:

W = \int \vec{F}\cdot d\vec{r}

In AP Physics C, you do not need to memorize every possible integral form, but you should be comfortable with the idea that “area under a force-versus-position graph” can represent work, and that only the component of force along the displacement contributes.

Kinetic energy in action: worked examples

Example 1: Using work to find speed (constant force)

A block of mass m = 2.0\ \text{kg} starts from rest on a frictionless horizontal surface. A constant horizontal force F = 6.0\ \text{N} acts over a distance d = 3.0\ \text{m}. Find the final speed.

Reasoning: If friction is absent, the net work is just the applied force’s work.

W_{\text{net}} = Fd = 6.0\cdot 3.0 = 18\ \text{J}

Work–kinetic energy theorem:

\Delta K = 18\ \text{J}

Since it starts from rest, K_i = 0, so

K_f = 18 = \frac{1}{2}mv^2 = \frac{1}{2}(2.0)v^2 = 1.0\,v^2

v^2 = 18

v = \sqrt{18} \approx 4.24\ \text{m/s}

Common pitfall: Plugging in velocity components incorrectly. Kinetic energy uses speed squared, not just one component (unless motion is purely along one axis).

Example 2: Force that varies with position (graph/area idea)

Suppose the net force on a particle along a line is F(x) = 4x (in newtons) from x = 0 to x = 2\ \text{m}. The work done is

W = \int_0^2 4x\,dx = 2x^2\bigg|_0^2 = 8\ \text{J}

That work equals \Delta K. Notice how energy methods let you bypass finding acceleration as a function of time.

Exam Focus

• Typical question patterns:
  • Compute a final speed using W_{\text{net}} = \Delta K without explicitly finding acceleration or time.
  • Use a force-versus-position graph to find work (area under the curve), then relate it to \Delta K.
  • Combine kinetic energy with potential energy via conservation to relate speeds at different positions.
• Common mistakes:
  • Treating kinetic energy as negative when velocity is negative. v^2 makes K nonnegative.
  • Forgetting that only net work changes K; a single force’s work is not necessarily \Delta K if other forces also act.
  • Mixing up displacement and distance in work problems when direction matters.

Potential Energy (Gravitational and Elastic)

What potential energy is

Potential energy is energy associated with the configuration of a system. The key word is “system”: potential energy is not usually owned by a single object in isolation, but by interacting objects (like Earth and a mass, or a spring and a block).

Potential energy is most useful when it is linked to a conservative force. For conservative forces, you can define a potential energy function U such that the work done by that force depends only on the initial and final positions, not on the path.

A crucial relationship is

W_{\text{cons}} = -\Delta U

This says: when a conservative force does positive work on the system, the system’s potential energy decreases, and vice versa.

Why potential energy matters

Potential energy lets you solve motion problems using positions instead of forces at every instant. Once you know how U changes between two points, you can connect that change to kinetic energy changes using energy conservation.

It also helps you reason qualitatively. For example, objects naturally move toward lower potential energy if only conservative forces do work (like a ball rolling down a hill without friction).

Gravitational potential energy

Near-Earth (uniform gravitational field approximation)

Close to Earth’s surface, gravity is approximately constant in magnitude and direction. The gravitational potential energy of an object-Earth system is

U_g = mgh

where h is height measured relative to a chosen zero level.

Two important clarifications:

• Only differences in potential energy matter physically. You can choose the zero of U_g wherever is convenient.
• The formula U_g = mgh is tied to the approximation of constant g and modest height changes compared to Earth’s radius.

The change in gravitational potential energy between heights h_i and h_f is

\Delta U_g = mg(h_f - h_i)

Universal gravitation (variable g)

When distances from Earth’s center vary significantly (orbital problems), gravitational potential energy is

U_g(r) = -\frac{GMm}{r}

where r is the distance between centers of mass, G is the gravitational constant, M is the mass of the attracting body (Earth, planet, etc.), and m is the smaller mass.

The negative sign is not a “mistake”—it encodes the convention that U \to 0 as r \to \infty. Being bound to a planet corresponds to negative gravitational potential energy.

Elastic (spring) potential energy

For an ideal spring that obeys Hooke’s law,

F_s = -kx

where k is the spring constant and x is displacement from equilibrium.

The associated elastic potential energy is

U_s = \frac{1}{2}kx^2

This expression is always nonnegative because it measures energy stored due to deformation. The equilibrium position is naturally the zero of spring potential energy when x = 0.

Notation reference (common equivalences)

Potential energy in action: worked examples

Example 1: Gravitational potential energy conversion (drop)

A ball of mass 0.50\ \text{kg} is dropped from rest from height h = 4.0\ \text{m} (ignore air resistance). Find its speed just before hitting the ground.

Concept: Gravitational potential energy converts to kinetic energy.

Choose the ground as U_g = 0. Initially:

U_i = mgh = 0.50\cdot 9.8\cdot 4.0 = 19.6\ \text{J}

Initially K_i = 0. At the ground, U_f = 0, so energy conservation (with only gravity doing work) gives

K_f = U_i = 19.6\ \text{J}

\frac{1}{2}mv^2 = 19.6

\frac{1}{2}(0.50)v^2 = 19.6

0.25v^2 = 19.6

v^2 = 78.4

v \approx 8.86\ \text{m/s}

Common pitfall: Putting h = 0 at the release point and then forgetting that \Delta U_g must still be computed consistently.

Example 2: Spring launch on a frictionless surface

A spring with k = 200\ \text{N/m} is compressed by x = 0.10\ \text{m} and launches a block of mass m = 0.50\ \text{kg} on a frictionless horizontal table. Find the block’s speed when it leaves the spring (at x = 0).

Initial energy is spring potential:

U_{s,i} = \frac{1}{2}kx^2 = \frac{1}{2}(200)(0.10)^2 = 1.0\ \text{J}

At release, spring potential is zero and the energy is kinetic:

\frac{1}{2}mv^2 = 1.0

\frac{1}{2}(0.50)v^2 = 1.0

0.25v^2 = 1.0

v^2 = 4.0

v = 2.0\ \text{m/s}

Common pitfall: Confusing x with the spring’s total length. x is the displacement from equilibrium length.

Exam Focus

• Typical question patterns:
  • Use U_g = mgh (or \Delta U_g = mg\Delta h) to relate speeds at different heights.
  • Use U_s = \frac{1}{2}kx^2 to connect spring compression/extension to speed.
  • Recognize when to use U_g = -GMm/r for orbital or large-height problems.
• Common mistakes:
  • Mixing up signs in W_{\text{cons}} = -\Delta U, especially with gravity.
  • Treating potential energy as belonging to a single object rather than an interacting system.
  • Using mgh when the problem context suggests variable gravitational field (large r changes).

Conservation of Energy

What conservation of energy means in mechanics

Conservation of energy is the idea that energy is not created or destroyed; it is transferred or transformed. In AP Physics C mechanics, you most often apply this through the conservation of mechanical energy, meaning kinetic plus potential energy:

E_{\text{mech}} = K + U

Mechanical energy is conserved when only conservative forces do work (or when non-conservative forces do zero net work).

So, if a system experiences only conservative forces:

K_i + U_i = K_f + U_f

This is not a new law separate from Newton’s laws—it’s a powerful consequence of how conservative forces behave.

Why this matters

Energy conservation is one of the fastest ways to solve AP Physics C problems because it:

• avoids time entirely,
• handles multi-stage motion cleanly,
• naturally combines multiple interactions (gravity plus spring, for example).

It also provides strong “sanity checks.” If your algebra predicts a negative value for v^2 in a situation that should be physically possible, it often means you mishandled a sign or included/excluded work incorrectly.

How to apply mechanical energy conservation step by step

A reliable process looks like this:

1. Define the system (what objects are included). This choice determines which forces are internal and which do work from outside.
2. Choose initial and final states (positions or configurations). Label them clearly.
3. Write energy expressions at each state: include relevant terms (kinetic, gravitational potential, spring potential). Don’t include terms that are constant or irrelevant.
4. Decide whether mechanical energy is conserved. If non-conservative forces (like kinetic friction) do work, mechanical energy changes.
5. Solve for the unknown (often speed, height, compression, etc.).

Mechanical energy with non-conservative work (bridge to the next section)

When non-conservative forces do work on the system, mechanical energy is not conserved. A commonly used relationship is

\Delta K + \Delta U = W_{\text{nc}}

where W_{\text{nc}} is the work done by non-conservative forces (like friction, air drag, an applied push if you treat it as external).

Equivalently,

\Delta E_{\text{mech}} = W_{\text{nc}}

This is extremely useful on AP problems involving friction: instead of tracking friction through Newton’s second law, you compute the work friction does and adjust the energy balance.

Conservation of energy in action: worked examples

Example 1: Pendulum speed at the bottom

A pendulum bob of mass m is released from rest at a vertical height difference h above its lowest point (ignore air resistance). Find the speed at the bottom.

At release:

K_i = 0

Choose the bottom as U_g = 0, so

U_i = mgh

At the bottom:

U_f = 0

K_f = \frac{1}{2}mv^2

Conservation:

mgh = \frac{1}{2}mv^2

Cancel m:

gh = \frac{1}{2}v^2

v = \sqrt{2gh}

Common pitfall: Forgetting that tension does no work on the bob (tension is perpendicular to motion along the arc), so it doesn’t appear in the energy equation.

Example 2: Block-spring-gravity combination

A block of mass m = 1.0\ \text{kg} slides down a frictionless track and compresses a spring at the bottom. The block starts from rest at height h = 0.80\ \text{m} above the spring’s uncompressed position. The spring has k = 400\ \text{N/m}. Find the maximum compression x.

Set states:

• Initial: at height h, spring uncompressed, speed zero.
• Final: at maximum compression, speed zero, height reference set to zero at spring level.

Initial energy:

K_i = 0

U_{g,i} = mgh = (1.0)(9.8)(0.80) = 7.84\ \text{J}

U_{s,i} = 0

Final energy:

K_f = 0

U_{g,f} = 0

U_{s,f} = \frac{1}{2}kx^2 = \frac{1}{2}(400)x^2 = 200x^2

Conservation:

7.84 = 200x^2

x^2 = 0.0392

x \approx 0.198\ \text{m}

Common pitfall: Including kinetic energy at maximum compression (it is zero there by definition of “maximum compression”).

Exam Focus

• Typical question patterns:
  • “Released from rest at height” problems where you solve for speed using K_i + U_i = K_f + U_f.
  • Multi-energy problems (gravity plus spring) where you identify the correct initial/final states.
  • Problems that test whether a force does work (normal force and tension often do zero work in constrained motion).
• Common mistakes:
  • Including forces in the energy equation instead of energies (energy methods track work/energy, not a force inventory).
  • Forgetting to include one type of potential energy (spring energy is commonly missed).
  • Assuming mechanical energy conservation even when friction or air resistance is present.

Conservative and Non-Conservative Forces

What “conservative” means (and why you care)

A conservative force is a force for which the work done between two points depends only on the initial and final positions—not on the path taken. This matters because it allows you to define a potential energy function U and use energy conservation in a clean way.

There are two equivalent characterizations you’ll see:

1. Path independence: Work from A to B is the same for any path.
2. Zero work over a closed loop: If you start at A, move around, and return to A, the total work is zero:

\oint \vec{F}\cdot d\vec{r} = 0

Gravity (idealized) and spring forces are classic conservative forces.

Conservative forces and potential energy

If a force is conservative, you can connect it to potential energy using

W_{\text{cons}} = -\Delta U

In one-dimensional motion along x, the force can be obtained from the potential energy function:

F_x = -\frac{dU}{dx}

This is conceptually powerful: the “shape” of a potential energy curve tells you how the force behaves. For instance, where U(x) slopes downward with increasing x, the force points in the +x direction (because of the minus sign).

Non-conservative forces

A non-conservative force is one for which the work done depends on the path. The most important examples in AP Mechanics are kinetic friction and air resistance (drag). These forces typically transform mechanical energy into thermal energy or other internal energy forms.

Because these forces do not have a well-defined potential energy function, you handle them using work terms:

\Delta K + \Delta U = W_{\text{nc}}

Interpretation: non-conservative work changes mechanical energy.

• If W_{\text{nc}} < 0 (like kinetic friction), mechanical energy decreases.
• If W_{\text{nc}} > 0 (like an external motor doing work), mechanical energy increases.

A practical system-choice viewpoint (how AP problems “want” you to think)

Many energy mistakes come from choosing an inconsistent system.

• If you choose block + Earth as your system, then gravity is internal and shows up as U_g.
• If you choose block only, then gravity is external and shows up through work W_g instead.

Both approaches can work, but you must not double-count by including U_g and also adding W_g as an external work term for the same interaction.

Comparing conservative vs non-conservative (useful on exams)

Forces that often confuse students (normal force, tension)

Normal force and tension are not automatically conservative or non-conservative in the way friction is. The key question is: do they do work in the situation?

• Normal force often does zero work when motion is along a surface because it is perpendicular to the displacement.
• Tension often does zero work in an ideal pendulum because it is perpendicular to the bob’s instantaneous displacement.

However, in other setups (like a rope pulling a block horizontally), tension can do nonzero work. Don’t memorize “tension does no work”—analyze geometry.

Non-conservative forces in action: worked examples

Example 1: Block sliding down with friction (energy with W_{\text{nc}})

A block of mass m = 3.0\ \text{kg} slides down a rough incline, dropping a vertical height of h = 2.0\ \text{m}. The kinetic friction force does work W_f = -18\ \text{J} on the block. The block starts from rest. Find the speed at the bottom.

Use

\Delta K + \Delta U = W_{\text{nc}}

Let the system be block + Earth, so gravitational potential energy is included. Choose the bottom as U_f = 0. Then

K_i = 0

U_i = mgh = (3.0)(9.8)(2.0) = 58.8\ \text{J}

U_f = 0

W_{\text{nc}} = -18\ \text{J}

Compute changes:

\Delta U = U_f - U_i = -58.8\ \text{J}

So

\Delta K - 58.8 = -18

\Delta K = 40.8\ \text{J}

Thus

K_f = 40.8 = \frac{1}{2}mv^2 = \frac{1}{2}(3.0)v^2 = 1.5v^2

v^2 = 27.2

v \approx 5.22\ \text{m/s}

Common pitfall: Using mgh = \frac{1}{2}mv^2 even though friction is present. Friction removes mechanical energy, so speed is lower than the frictionless case.

Example 2: Checking whether a force is conservative (conceptual)

Suppose you push a box across the floor from point A to B.

• Gravity and the normal force do no net work (displacement is horizontal).
• Kinetic friction does negative work that depends on how far you slid the box. If you take a longer path from A to B, friction does more negative work.

That dependence on path length is the signature of a non-conservative force.

Memory aid for sign and interpretation

A simple way to remember the relationship between conservative work and potential energy is:

• “Down in potential means positive conservative work.”

Because

W_{\text{cons}} = -\Delta U

If \Delta U is negative, then W_{\text{cons}} is positive.

Exam Focus

• Typical question patterns:
  • Decide whether mechanical energy is conserved; if not, incorporate friction via W_{\text{nc}}.
  • Use a potential energy function U(x) (or graph) to infer force direction/magnitude via slope ideas.
  • Identify which forces do work (especially normal force, tension) based on geometry.
• Common mistakes:
  • Double-counting gravity by including both U_g and W_g in the same equation.
  • Sign errors with friction work: kinetic friction typically makes W_{\text{nc}} negative.
  • Assuming “conservative” means “constant.” A force can vary with position and still be conservative (spring force is the classic example).