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3.1 Translational Kinetic Energy

Connect motion to energy

An object has translational kinetic energy because its center of mass is moving. For an object of mass $m$ moving with velocity $\vec v$ , use the speed $v=|\vec v|$ in the kinetic-energy equation.

$\begin{gathered}K=\frac{1}{2}mv^2\\ [K]=\text{J}=\text{kg}\,\text{m}^2/\text{s}^2\end{gathered}$

Read the squared-speed relationship

At fixed speed, doubling the mass doubles $K$ . At fixed mass, doubling the speed makes $K$ four times as large because velocity magnitude is squared. Reversing the direction of motion without changing speed leaves $K$ unchanged.

Calculate with units

Worked example: A $2.0\,\text{kg}$ cart moves at $3.0\,\text{m/s}$ relative to the laboratory.

$K=\dfrac12(2.0\,\text{kg})(3.0\,\text{m/s})^2=9.0\,\text{J}$ .

The cart therefore has $9.0\,\text{J}$ of translational kinetic energy in the laboratory frame.

State whose measurement it is

Kinetic energy is a scalar, so it has no direction, but its value depends on the observer's reference frame. An observer moving alongside the cart at $3.0\,\text{m/s}$ measures the cart's speed as zero and therefore measures $K=0$ , while the laboratory observer measures $9.0\,\text{J}$ . Both values are correct in their own frames.

3.2 Work

Identify the energy transfer

Work is energy transferred into or out of a system when a force acts through a displacement of its point of application. Work is a scalar: positive work transfers energy into the system, negative work transfers energy out, and zero work transfers no energy by that force.

$W=F_{\parallel}d=Fd\cos\theta$

Use the parallel force

Constant-force example: A $20\,\text{N}$ force acts $60^\circ$ to a cart's $3.0\,\text{m}$ displacement.

$W=(20\,\text{N})(3.0\,\text{m})\cos60^\circ=30\,\text{J}$ .

Only the parallel component, $F_{\parallel}=10\,\text{N}$ , transfers energy. A perpendicular component can turn the velocity but does zero work and does not change kinetic energy.

Connect net work to kinetic energy

For all forces acting on an object, add their work algebraically:

$\Delta K=K_f-K_i=\sum_i W_i=W_{\text{net}}.$

Thus positive net work increases kinetic energy, negative net work decreases it, and zero net work leaves it unchanged. When the system's center of mass and the force's point of application move the same distance, the system may be modeled as an object for this kinetic-energy change.

Read work from a graph

On a graph of the parallel force component $F_{\parallel}$ versus displacement $x$ , work equals the signed area between the curve and the $x$ -axis. Area above the axis is positive work; area below it is negative work. This rule also handles a force that changes with position.

Distinguish force types

Force type	Does work depend on the path?	Consequence
Conservative	No—only the initial and final configurations matter	Potential energy can be associated with the interaction; returning to the initial configuration gives zero total work by that force
Nonconservative, such as friction or air resistance	Yes	Mechanical energy may be dissipated as thermal energy or sound; for constant friction opposing motion, $W_f=-F_fd$ and the dissipated amount is $F_fd$

Control the system model

Always name the system and the displacement of the force's point of application. An external force can also change a system's internal configuration, so its work need not appear only as center-of-mass kinetic energy. AP Physics 1 analyzes mechanical-energy transfer here; it recognizes dissipation to thermal energy or sound but does not require the AP Physics 2 treatment of energy transfer by heating or cooling.

3.3 Potential Energy

Name the interacting system

Potential energy is a scalar associated with the relative positions of objects in a system that interact through conservative forces. It belongs to the interacting system—not to one isolated object. A mass–spring system can store elastic potential energy; a mass–planet system can have gravitational potential energy.

Choose a consistent zero

The observer chooses where $U=0$ to simplify analysis. That choice changes the numerical value of $U$ , but not a physically meaningful change $\Delta U=U_f-U_i$ when one consistent reference is used.

Match the model to its conditions

System and condition	Potential-energy relationship
Ideal spring displaced $\Delta x$ from equilibrium	$U_s=\tfrac12k(\Delta x)^2$
Two approximately spherical masses separated center-to-center by $r$	$U_g=-G\dfrac{m_1m_2}{r}$
Object–planet system near a surface where $g$ is nearly constant	$\Delta U_g=mg\Delta y$

Calculate a change

Near-surface example: A $2.0\,\text{kg}$ object rises $3.0\,\text{m}$ where $g=9.8\,\text{N/kg}$ .

$\Delta U_g=(2.0\,\text{kg})(9.8\,\text{N/kg})(+3.0\,\text{m})=+59\,\text{J}$ .

The object–Earth system gains $59\,\text{J}$ of gravitational potential energy. The result is the same whichever height was labeled zero.

Sum pairs and respect limits

For a system with more than two objects, total potential energy is the sum of the potential energy of each interacting pair. Do not add a separate potential energy for a lone object, and do not mix model conditions: $mg\Delta y$ is a near-surface approximation, whereas $-Gm_1m_2/r$ is the general two-spherical-mass form.

3.4 Conservation of Energy

Start with the boundary

Before listing energy, draw or state the system boundary. Kinetic energy can belong to a moving object. Potential energy belongs to a system with an internal conservative interaction or reversible change of shape.

Classify the system

Chosen system	Energies it may contain	Why
One object only	Kinetic energy	No second object or reversible internal interaction is included to store potential energy
Interacting objects, such as object–Earth	Kinetic and gravitational potential energy	The conservative gravitational interaction is inside the system
Cart–ideal-spring system	Kinetic and elastic potential energy	Motion changes a reversible spring deformation inside the system

Assign potential energy correctly

A raised object alone does not contain gravitational potential energy; that energy belongs to the object–Earth system. Including several objects is not sufficient by itself—the system needs the relevant conservative interaction or reversible deformation to have potential energy.

Build the mechanical-energy account

A system's mechanical energy is the sum of all kinetic and potential energies included in that system.

$\begin{gathered}E_{\text{mech}}=K+U\\ \Delta E_{\text{system}}=E_{\text{transferred into system}}-E_{\text{transferred out}}\end{gathered}$

Track an internal conversion

For an object–Earth system falling without air resistance, gravitational potential energy decreases while kinetic energy increases by the same amount:

$\Delta K=-\Delta U_g,\qquad K_i+U_i=K_f+U_f.$

Energy changes form within the system, so its total mechanical energy remains constant.

Interpret conservation correctly

Conservation does not mean every energy type stays constant. One type can decrease as another increases. If the system's total energy changes, an equal amount of energy must have crossed between the system and its surroundings.

Keep total energy conserved

Energy is conserved in every interaction, but the energy assigned to a selected system can change when energy crosses its boundary. The same event can therefore have different—but consistent—energy accounts for different system choices.

Analyze one event two ways

Same falling event	What is inside?	Energy account
System = object only	The object, but not Earth	Gravity is external and does positive work; the object's kinetic energy increases
System = object + Earth	Object and conservative gravitational interaction	Gravity is internal; $U_g$ decreases while $K$ increases, with no energy transfer required across the boundary

Test mechanical-energy constancy

The selected system's total mechanical energy is constant when both conditions hold: (1) work done on the system from outside is zero, and (2) there are no nonconservative interactions within the system. If external work is nonzero, that work transfers energy between the system and environment.

Account for dissipation

With friction or air resistance, mechanical energy can be dissipated as thermal energy or sound. Mechanical energy may decrease, but energy itself is not destroyed; a complete account includes the transferred or transformed energy.

3.5 Power

Read power as a rate

Power is the rate at which energy is transferred into or out of a system, or converted from one form to another within it. The SI unit is the watt: $1\,\text{W}=1\,\text{J/s}$ .

$P_{\text{avg}}=\frac{\Delta E}{\Delta t}=\frac{W}{\Delta t}$

Calculate an interval average

Worked example: A force transfers $600\,\text{J}$ of energy to a system in $3.0\,\text{s}$ .

$P_{\text{avg}}=\dfrac{600\,\text{J}}{3.0\,\text{s}}=200\,\text{W}$ .

On average, energy entered the system at $200\,\text{J}$ each second during that interval.

Connect force and velocity

At one instant, a constant force delivers power according to

$P_{\text{inst}}=F_{\parallel}v=Fv\cos\theta,$

where $\theta$ is the angle between force and velocity. A perpendicular force has $F_{\parallel}=0$ and delivers zero instantaneous power, even if it changes the direction of motion.

Separate power from energy

Power and energy are not interchangeable. A high-power transfer can move a given amount of energy in less time; it does not necessarily transfer more total energy unless the time interval is also known. Average power describes an interval, while instantaneous power describes one moment.

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