Unit 3: Work, Energy, and Power

Work Done by a Constant Force

Definition and Directionality

• Work is a measure of energy transferred to or from an object via a force causing displacement. It is defined as \( W = \vec{F} \cdot \vec{d} = Fd \cos \theta \), where \( \theta \) is the angle between the force and displacement vectors. If the force acts in the direction of displacement, work is positive; if it opposes the motion, work is negative. Work is a scalar quantity but can have a positive or negative sign to indicate energy input or removal. This concept is essential when analyzing systems involving ramps, friction, or tension doing work over distance.
• If no displacement occurs, no work is done regardless of how large the force is. For example, pushing against a wall with 500 N of force that doesn’t move results in zero work, because \( d = 0 \). This highlights the difference between energy exertion (biological effort) and physical work (mechanical energy transfer). The work equation depends critically on both displacement and the component of force aligned with that motion. Understanding this helps avoid confusion between effort and mechanical energy transfer in real-world systems.
• When the force is not constant — such as when a spring is stretched or a rocket's thrust changes — the total work done must be found using a definite integral: \[ W = \int_{x_i}^{x_f} F(x)\,dx \] This integral-based definition of work becomes especially useful in AP Physics C for problems involving variable forces. It provides a direct connection between calculus and physics by showing how work is accumulated over continuous motion. We’ll return to this formulation when analyzing spring energy, resistive forces, and conservative fields in later sections.

Kinetic Energy and the Work-Energy Theorem

Linking Force to Motion Through Energy

• Kinetic energy is the energy of motion and is given by the expression \( K = \frac{1}{2}mv^2 \). This quantity increases with both mass and the square of velocity, meaning doubling the speed quadruples the kinetic energy. Kinetic energy is always a positive scalar and represents the capacity of a moving object to do work on another object. Unlike momentum, which is a vector, kinetic energy is purely scalar and additive across systems. It plays a central role in energy conservation and impact problems.
• The Work-Energy Theorem states that the net work done on an object is equal to the change in its kinetic energy: \[ W_{\text{net}} = \Delta K = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 \] This theorem provides an alternative to kinematics when solving for speed or displacement under variable or unknown acceleration. Rather than tracking forces over time, energy analysis simplifies motion by relating changes in velocity directly to net work. It’s especially powerful in problems involving ramps, tension, friction, or springs where forces may vary or act indirectly.
• Because the Work-Energy Theorem includes net work, all forces must be considered — including friction, normal force (if it does work), tension, or applied forces. If conservative forces like gravity or springs are the only contributors, then the work done by those forces can be equated directly to energy changes. However, if non-conservative forces are present, they remove or add energy and must be factored into the net work calculation. This comprehensive approach prepares students for complex energy tracking in future units such as momentum, rotation, and oscillations.

Potential Energy

Gravitational and Elastic Energy Storage

• Gravitational potential energy is energy stored due to an object’s position in a gravitational field. Near Earth’s surface, it is calculated as \( U_g = mgh \), where \( h \) is the vertical height relative to a reference level. The choice of zero height is arbitrary, but consistent selection is important for meaningful energy comparisons. Changes in potential energy correspond to work done by gravity, which can be positive or negative depending on the direction of motion. This energy transforms into kinetic energy as objects fall and plays a major role in conservation of energy problems.
• Elastic potential energy is the energy stored in a compressed or stretched spring, given by the formula: \[ U_s = \frac{1}{2}kx^2 \] where \( k \) is the spring constant and \( x \) is the displacement from equilibrium. This form of potential energy is conservative and perfectly reversible in ideal (frictionless) systems. It arises in systems involving mass-spring oscillators, bowstrings, trampolines, and elastic collisions. The graph of \( F \) vs. \( x \) for a spring is linear, and the area under it represents the energy stored — highlighting the connection between force and energy through calculus.
• Potential energy is a form of “stored work” and is recovered as kinetic energy when the system is released. While kinetic energy depends on velocity, potential energy depends on configuration (height, compression, etc.), and both contribute to the system’s total mechanical energy. The ability to convert energy between potential and kinetic forms underlies motion in roller coasters, pendulums, and orbital systems. Later, this idea will connect to conservation of energy in rotating and oscillating systems, where the interplay of potential and kinetic energy defines periodic behavior. Understanding potential energy is essential for recognizing energy flows within systems, especially when external forces are absent.

Conservation of Mechanical Energy

Ideal Systems with No Energy Loss

• In the absence of non-conservative forces like friction or air resistance, the total mechanical energy (TME) of a system remains constant: \[ E_{\text{mech}} = K + U = \text{constant} \] This means kinetic and potential energy can transform into one another, but their sum does not change. For example, an object falling under gravity loses potential energy and gains kinetic energy by the same amount. This principle allows students to solve problems using energy comparisons without needing to calculate acceleration, time, or net force.
• Conservation of energy is especially powerful for analyzing motion on hills, ramps, or pendulum arcs. If an object starts at rest with known height, you can directly calculate speed at the bottom using energy principles. These problems emphasize energy transfer over time-independent motion, allowing for elegant solutions to complex trajectories. However, it’s important to verify the absence of friction or other energy losses before assuming conservation. On the AP exam, recognizing when conservation applies is just as important as being able to use it mathematically.
• In more advanced cases, students must track potential energy from both gravity and springs. In such problems, mechanical energy conservation includes multiple types of potential energy, and the total is written as: \[ \frac{1}{2}mv_i^2 + \frac{1}{2}kx_i^2 + mgh_i = \frac{1}{2}mv_f^2 + \frac{1}{2}kx_f^2 + mgh_f \] This formulation becomes essential in oscillation and orbital motion problems, where energy is constantly being exchanged between potential and kinetic forms. It also reinforces the importance of defining a consistent zero-point for both spring length and gravitational height.

Non-Conservative Work and Thermal Energy

Energy Loss Due to Friction or Drag

• When non-conservative forces like friction or air resistance are present, mechanical energy is no longer conserved. These forces convert mechanical energy into thermal energy or sound, which is not recoverable within the system. The total energy is still conserved (as required by the law of conservation of energy), but it now includes internal energy increases. The work done by non-conservative forces can be written as: \[ W_{\text{nc}} = \Delta E_{\text{mech}} = \Delta K + \Delta U \] This equation allows students to calculate how much energy is lost from the useful mechanical system due to dissipative forces.
• Frictional work can be calculated using \( W = -f_k d \), where the negative sign indicates energy is being removed from the system. This value should be included in energy equations as a loss when solving for final speed, height, or stretch. These problems usually involve sliding blocks, ramps, or dragging across rough surfaces, and help students understand how energy is not always conserved in usable form. Being able to distinguish conservative and non-conservative forces is essential for interpreting energy bar charts and solving real-world scenarios. The AP exam often tests whether students correctly apply energy conservation only when appropriate.
• Including non-conservative work expands the energy model to match reality more closely. For example, in a system where a box slides down a ramp with friction, students must calculate both the gravitational energy lost and the amount dissipated as heat. This allows them to determine how much kinetic energy the box has at the bottom. These problems also help transition to thermodynamics and systems thinking in more advanced physics. Students should always start energy analyses by identifying all the forces doing work and whether each is conservative or non-conservative.

Power

Rate of Energy Transfer

• Power is defined as the rate at which work is done or energy is transferred: \[ P = \frac{W}{t} \] It is measured in watts (W), where 1 W = 1 J/s. Power gives us insight into how quickly a machine or system can deliver energy, rather than just how much total work it performs. For example, lifting a box slowly or quickly requires the same amount of work, but more power is required to do it quickly. Understanding power is critical in systems where time matters — such as engines, motors, and energy consumption.
• When force and velocity are involved simultaneously, power can also be written as: \[ P = \vec{F} \cdot \vec{v} \] This version shows that instantaneous power depends on both the magnitude of force and the velocity of the object. It becomes especially useful when analyzing constant-speed motion with opposing forces like friction or drag. Students must ensure that the direction of force and velocity is accounted for, since this equation also includes a dot product. In rotating systems, this concept will later extend to torque and angular velocity.
• Power also connects to energy efficiency and performance in real-world systems. In practical contexts, machines and engines are often rated by horsepower (1 hp = 746 W) or kilowatts. A system that uses more power to do the same amount of work in less time is said to be more "powerful" but not necessarily more energy-efficient. Students should recognize that while energy is conserved, power tells us how fast that energy is being delivered. On the AP exam, problems involving elevators, motors, or ramps often test this idea by combining force, speed, and time.

Energy Transfer in Isolated and Non-Isolated Systems

Understanding ΔE = ΣT and When Mechanical Energy Is Conserved

• According to the law of conservation of energy, energy cannot be created or destroyed — only transformed or transferred. In a non-isolated system, energy can enter or leave the system due to external forces doing work, and the change in the system’s total energy is equal to the sum of the energy transferred: \[ \Delta E_{\text{system}} = \sum T \] This includes both mechanical and internal energy changes, depending on how forces act within the system. Recognizing whether a system is isolated or not determines whether you should account for external work or internal losses due to non-conservative forces.
• If the system is isolated, then no energy is transferred into or out of it — meaning: \[ \Delta E_{\text{system}} = 0 \] In such cases, the only allowable changes occur internally, and the total energy remains constant. The internal energy may still change due to friction or other dissipative forces, but no energy crosses the system’s boundary. This forms the basis for the conservation of mechanical energy when friction and other non-conservative forces are absent.
• The internal energy of a system changes due to the work done by non-conservative forces like friction. These forces transform useful mechanical energy into thermal energy, which increases the internal energy of the system: \[ \Delta E_{\text{internal}} = -W_{\text{nc}} \] A non-conservative force is path-dependent, meaning the total work it does depends on the distance traveled, not just the initial and final positions. Conservative forces, like gravity and springs, are path-independent and allow for full recovery of energy. This distinction is critical for solving energy problems involving ramps, loops, and drag.
• Because most non-conservative forces in mechanics are friction-related, we can often write: \[ W_{\text{friction}} = \Delta ME \] where \( \Delta ME = ME_f - ME_i \) is the change in mechanical energy. This is valid only when friction is the only non-conservative force doing work and no energy enters or exits the system externally. The equation shows that the mechanical energy lost due to friction becomes internal energy (often heat), and this loss is irreversible within the system.
• If the system is both isolated and there is no friction, then we have true conservation of mechanical energy: \[ \Delta ME = 0 \Rightarrow ME_i = ME_f \] This means kinetic and potential energy may transform into each other, but their sum stays constant. This condition must be explicitly verified in every energy problem — students should not assume mechanical energy is conserved unless the problem clearly states there’s no friction or external work. This principle underlies many idealized AP Physics questions involving springs, ramps, or projectiles.
• Whenever you use an energy equation like \( W_{\text{friction}} = \Delta ME \) or \( ME_i = ME_f \), it is essential to clearly define your energy reference points. This includes identifying the object’s initial and final positions and the horizontal zero line for gravitational potential energy. Misidentifying these reference points leads to incorrect energy differences and invalid conclusions. Consistency and clarity in defining potential energy baselines are just as important as the math itself.

Forces and Potential Energy

Using Derivatives to Relate Force and Potential Energy

• In AP Physics C, force and potential energy are directly connected through calculus. Specifically, the force exerted by a conservative field is the negative derivative of the potential energy function: \[ F(x) = -\frac{dU}{dx} \] This equation means that force always acts in the direction that reduces potential energy. Graphically, if the slope of the potential energy curve is positive, the force is negative (leftward), and if the slope is negative, the force is positive (rightward). This connection allows us to analyze motion using energy landscapes even without knowing the full equations of motion.
• This derivative relationship applies to all conservative forces — gravity, springs, electric fields — and helps visualize stability. At a minimum in potential energy (where \( \frac{dU}{dx} = 0 \)), the force is zero and the system is in equilibrium. If the point is a local minimum, it’s stable (object oscillates around it); if it's a maximum, the point is unstable (object accelerates away). This idea will be especially useful in oscillations and potential wells in Unit 7 and in electric/magnetic systems in Physics C: E&M.
• You can also derive potential energy functions by integrating force: \[ U(x) = -\int F(x)\,dx + C \] where \( C \) is an arbitrary constant based on your choice of reference level. This approach is commonly used to derive spring potential energy from Hooke’s Law or to model custom force fields. By interpreting potential energy curves, students can predict motion behavior — including turning points, equilibrium, and regions of acceleration — without solving differential equations. Mastery of this concept makes energy analysis more powerful than force diagrams in many cases.

Power (Extended Concepts)

Average vs. Instantaneous Power and Force-Velocity Link

• Average power is the total work done divided by the time interval: \[ P_{\text{avg}} = \frac{W}{\Delta t} \] It tells you the rate at which energy is transferred or transformed over a process. For example, a motor that does 600 J of work in 3 seconds produces 200 W of average power. This concept is useful in comparing systems that perform the same amount of work at different speeds, such as cars accelerating over varying distances.
• Instantaneous power is the power output at a specific moment in time and is defined by: \[ P = \vec{F} \cdot \vec{v} \] This dot product means power depends on the component of velocity in the direction of the force. If the force and velocity are in the same direction, power is positive; if opposite, power is negative (energy is being removed). This version is especially helpful in dynamic problems where both velocity and force vary over time, such as a rocket accelerating or friction opposing motion.
• Power also gives insight into real-world performance and efficiency. High-power systems can do the same work in less time but may also waste more energy as heat. For example, two motors may lift the same object to the same height, but the more powerful one does it faster — though not necessarily more efficiently. Students should distinguish between total energy (work) and the rate of transfer (power), especially in AP questions involving engines, elevators, or resistive forces. In later units, rotational analogs like torque and angular velocity are used in power calculations for spinning systems.

Work as an Integral of Force

Generalized Work for Variable Forces

• In calculus-based physics, work is defined as the definite integral of the force component in the direction of motion over a displacement: \[ W = \int_{x_i}^{x_f} F(x)\,dx \] This formulation is essential when the force is not constant, such as a spring force, gravitational field that varies with distance, or any force given by a function of position. The integral computes the area under the force–displacement curve, representing the total energy transferred to or from the object. This equation allows you to solve for net work even when force changes with location, something algebra-based physics cannot handle.
• For example, consider a spring force \( F(x) = -kx \). The work done by the spring as it moves from \( x_i \) to \( x_f \) is: \[ W = \int_{x_i}^{x_f} (-kx)\,dx = -\frac{1}{2}k(x_f^2 - x_i^2) \] This shows that energy is recovered or stored depending on whether the spring is compressing or expanding. This exact integral also leads directly to the definition of elastic potential energy: \[ U_s = \frac{1}{2}kx^2 \] Mastering these integrals helps students understand where energy storage equations come from, rather than memorizing them.
• When force is a nonlinear function (e.g., \( F(x) = ax^3 \)), the same integration rules apply: \[ W = \int_{x_i}^{x_f} ax^3\,dx = \frac{a}{4}(x_f^4 - x_i^4) \] This approach generalizes work to any system where the force varies with distance. These problems often appear on AP free-response sections, and students fluent in calculus can solve them quickly and accurately. The key is always identifying the force function and applying the limits of integration based on the object's displacement.

Force from Potential Energy Using Derivatives

The Gradient of Potential Energy

• When a potential energy function is known, the force at any position can be found using the derivative: \[ F(x) = -\frac{dU}{dx} \] This equation tells us that force always points in the direction of decreasing potential energy. This relationship applies to all conservative forces, including springs, gravity, and electrostatics. In physics problems, being able to take this derivative allows you to find turning points, equilibrium positions, and the direction of acceleration.
• If the second derivative of \( U(x) \) is positive at a point, that point is a local minimum — which means it’s a stable equilibrium. If the second derivative is negative, it’s a local maximum and represents an unstable equilibrium. These concepts allow students to classify motion behavior around potential wells using calculus, without needing to simulate the motion itself. Understanding this connection is critical for later topics like oscillations, molecular bonding, and orbital mechanics.
• For example, if \( U(x) = 3x^2 - x^4 \), then: \[ F(x) = -\frac{dU}{dx} = - (6x - 4x^3) = 4x^3 - 6x \] Solving \( F(x) = 0 \) gives equilibrium points. Then taking the second derivative of \( U \) lets you classify those points as stable or unstable. This type of analysis is often tested in AP conceptual questions where interpreting graphs of \( U(x) \) and \( F(x) \) is required.

Instantaneous Power from Calculus

Defining Power as the Derivative of Work

• Instantaneous power is the derivative of work with respect to time: \[ P(t) = \frac{dW}{dt} \] Since work is the integral of force over distance, we can apply the chain rule to express power in terms of force and velocity: \[ P(t) = \frac{dW}{dt} = \frac{d}{dt} \left( \int \vec{F} \cdot d\vec{r} \right ) = \vec{F} \cdot \vec{v} \] This form is extremely useful when the velocity and force are known functions of time or space. It lets you compute power even when speed is changing or direction is not constant.
• This equation also works in one-dimensional systems, where: \[ P(t) = F(t) \cdot v(t) \] For example, if \( F(t) = 6t \) N and \( v(t) = 3t^2 \) m/s, then: \[ P(t) = 6t \cdot 3t^2 = 18t^3 \] Integrating this gives the total work over time. This tight connection between work, force, and power helps students navigate real-world problems where engines, machines, or resistive forces vary with time.
• Understanding power as a derivative also helps reinforce the idea of energy rate. Just like velocity is the rate of change of position and acceleration is the rate of change of velocity, power is the rate at which energy is being used or delivered. This viewpoint becomes especially useful when dealing with energy flow in circuits, rotating systems, or even biological systems. The AP exam may test this concept by giving a time-dependent force and asking students to derive expressions for power or energy.

Example Problem 1: Force from a Potential Energy Graph

Problem: An object of mass 2.0 kg moves along the x-axis under the influence of a conservative force. The potential energy associated with this force is given by \( U(x) = 4x^2 - 2x \), where \( U \) is in joules and \( x \) is in meters. Find:

• (a) The expression for the force as a function of position
• (b) The equilibrium points and their stability
• (c) The speed of the object when it moves from \( x = 0.5\,\text{m} \) to \( x = 0.2\,\text{m} \)

Solution:

(a) Use \( F(x) = -\frac{dU}{dx} \):

\[ F(x) = -\frac{d}{dx}(4x^2 - 2x) = -(8x - 2) \Rightarrow F(x) = 2 - 8x \]

(b) Set \( F(x) = 0 \Rightarrow 2 - 8x = 0 \Rightarrow x = 0.25\,\text{m} \)

To determine stability, take the second derivative: \( \frac{d^2U}{dx^2} = 8 > 0 \), so this is a stable equilibrium.

(c) Use conservation of mechanical energy:

\[ K_i + U_i = K_f + U_f \quad \text{(starts from rest)} \Rightarrow 0 + U(0.5) = \frac{1}{2}mv^2 + U(0.2) \] \[ U(0.5) = 4(0.5)^2 - 2(0.5) = 1 - 1 = 0 \\ U(0.2) = 4(0.04) - 2(0.2) = 0.16 - 0.4 = -0.24 \] \[ 0 = \frac{1}{2}(2)v^2 - 0.24 \Rightarrow v^2 = 0.24 \Rightarrow v = \sqrt{0.24} ≈ 0.49\,\text{m/s} \]

Example Problem 2: Energy Dissipated by Friction

Problem: A 3.0 kg box is pushed with an initial speed of 4.0 m/s along a horizontal surface with kinetic friction. The coefficient of kinetic friction is \( \mu_k = 0.2 \). Determine:

• (a) The distance the box slides before coming to rest
• (b) The amount of energy converted into thermal energy
• (c) Confirm that energy is conserved when accounting for friction

Solution:

(a) Use work–energy theorem: \( W_{\text{net}} = \Delta K \)

Friction does negative work: \( W_f = -f_k d = -\mu_k mg d \)

\[ \Delta K = 0 - \frac{1}{2}mv^2 = -\frac{1}{2}(3)(4)^2 = -24\,\text{J} \] \[ -\mu_k mg d = -24 \Rightarrow d = \frac{24}{(0.2)(3)(9.8)} ≈ 4.08\,\text{m} \]

(b) Thermal energy generated is the same as energy lost: 24 J

(c) Mechanical energy decreases by 24 J, but that energy is not lost — it has been converted into heat via friction. So total energy is conserved when we include internal energy increase. This validates the principle of conservation of energy in systems with non-conservative forces.

Example Problem 3: Instantaneous Power from Force and Velocity

Problem: A car of mass 1000 kg is traveling at a constant speed of 20 m/s against a resistive force of 500 N (air drag and rolling resistance). The engine provides a constant force to counter this resistance. Calculate:

• (a) The instantaneous power output of the engine
• (b) How much work the engine does over 10 seconds
• (c) The energy wasted as heat due to resistive forces in that time

Solution:

(a) Use: \[ P = \vec{F} \cdot \vec{v} = (500)(20) = 10{,}000\,\text{W} = 10\,\text{kW} \]

(b) Work is force × distance, and distance = speed × time = 20 × 10 = 200 m

\[ W = F \cdot d = 500 \cdot 200 = 100{,}000\,\text{J} \]

(c) Since all of the engine’s output goes into overcoming friction (no acceleration), the entire 100,000 J is dissipated as heat. This is a clear example of non-conservative forces converting mechanical energy into internal energy while maintaining total energy conservation.

Common Misconceptions:

Clarifying Errors with Energy Concepts and Calculus

• “Work is always force times distance.”
  This formula only applies when the force is constant and acts in the direction of motion. When the force varies with position, you must use the integral \( W = \int F(x)\,dx \). Students who blindly use \( W = Fd \) may get incorrect results when dealing with springs, position-dependent forces, or curved potential energy graphs. Recognizing when force is a function of \( x \) is key to applying the correct calculus-based definition. The AP exam often includes variable-force problems specifically to test this distinction.
• “If potential energy increases, force must also increase.”
  Students often confuse potential energy and force, assuming a higher potential energy means a larger force. However, force is given by the negative slope of the potential energy function: \[ F(x) = -\frac{dU}{dx} \] So even if \( U(x) \) is increasing, the force might be negative, zero, or changing direction depending on the rate of change. Understanding that force depends on the derivative — not the height — of the potential energy curve is crucial for interpreting graphs and solving motion problems.
• “If work is positive, kinetic energy always increases.”
  This is only true if no other forces are doing negative work at the same time. In real problems, multiple forces may act — some adding energy, others removing it. The correct interpretation comes from the Work-Energy Theorem: \[ W_{\text{net}} = \Delta K \] Students must distinguish between individual force contributions and the net work, especially in systems involving friction, tension, or multiple interacting objects. Misunderstanding this often leads to incorrect final speeds or directions.
• “Power is just energy divided by time, even if the force changes.”
  While average power can be calculated as \( P = \frac{W}{t} \), this formula doesn’t account for changing force or velocity. In cases with time-varying forces or speeds, you must use: \[ P = \vec{F}(t) \cdot \vec{v}(t) \] Students who use average formulas when the situation calls for instantaneous power will miss how power evolves over time. The AP exam may test this by giving you \( F(t) \) and asking you to derive or interpret \( P(t) \).
• “If total mechanical energy is conserved, there are no forces doing work.”
  This is false — conservative forces like gravity or spring force still do work, but they don’t change the total mechanical energy. Instead, they convert energy from one form to another (e.g., potential to kinetic). Students often think conservation means “no forces acting,” which contradicts the concept of internal energy transfers within a system. Clarifying that **conservative work causes internal energy transformations**, not external energy gain/loss, is essential for solving motion problems involving ramps, springs, or pendulums.