Unit 3: Work, Energy, and Power

Work

In physics, work is defined as the transfer of energy to or from an object via a force that causes displacement. Work is not just about applying a force — for work to be done, the object must move, and the force must have a component in the direction of the displacement. If you push on a wall and it doesn’t move, no work is done in the physics sense, even though you may become tired. This is because work is mathematically tied to displacement, which is zero in this case. Work is measured in joules (J), where one joule equals one newton of force applied through one meter of displacement.

\[ W = Fd \cos \theta \]

Key Ideas about Work

• Work depends on the magnitude of the force, the displacement of the object, and the angle between them. If the force is parallel to displacement, the cosine term equals 1, and the full force contributes to the work. If the force is perpendicular (such as centripetal force in uniform circular motion), no work is done because the force does not change the object’s speed.
• Positive work occurs when the force component is in the same direction as the object’s displacement. For example, when you push a shopping cart forward, the work you do increases the cart’s kinetic energy. Negative work occurs when the force opposes the displacement, such as friction slowing a moving object, which decreases its kinetic energy.
• Work is a scalar quantity, meaning it has magnitude but no direction. However, the sign of the work (+ or –) indicates whether energy is being added to or taken away from the system. This makes it a powerful tool for analyzing changes in a system’s energy state.
• Work done by conservative forces like gravity is independent of the path taken. This means lifting an object straight up or moving it along a sloped ramp requires the same work against gravity if the height gained is the same. In contrast, nonconservative forces like friction depend on the path, so longer distances produce more energy loss.
• In AP Physics 1 problems, always check whether the force has a component in the displacement direction. Many students mistakenly calculate work using the full force, even if only part of the force contributes to moving the object. Drawing a free-body diagram helps avoid this mistake.

Translational Kinetic Energy

Kinetic energy is the energy an object has due to its motion. For straight-line (translational) motion, kinetic energy is expressed mathematically as:

\[ KE = \frac{1}{2}mv^2 \]

• Kinetic energy depends on both mass and velocity. Because velocity is squared, speed has a much greater effect than mass. For example, doubling the velocity of a car quadruples its kinetic energy, which is why higher speeds lead to much greater stopping distances.
• Kinetic energy is always a positive value or zero. An object at rest has zero kinetic energy, while any moving object has a positive amount. There is no such thing as negative kinetic energy, although negative work can reduce an object’s kinetic energy.
• The unit of kinetic energy is the joule (J), the same unit as work. This connection is not a coincidence — the work-energy theorem shows that the net work done on an object equals the change in its kinetic energy. Thus, whenever work is performed, kinetic energy changes accordingly.
• Kinetic energy is a scalar quantity, so it has no direction. However, changes in direction can still affect an object’s kinetic energy if the speed changes. For instance, in uniform circular motion, the speed remains constant, so kinetic energy does not change, even though velocity direction is changing.
• Real-world examples of kinetic energy include cars traveling on highways, athletes sprinting in races, and raindrops falling toward the Earth. Understanding kinetic energy allows physicists to predict the motion of objects and engineers to design systems such as brakes, roller coasters, and airbags that manage energy safely.

Work at an Angle

The formula for work must account for cases where the applied force is not parallel to the displacement. When a force is applied at an angle, only the component of the force that acts in the direction of the displacement contributes to the work. This is why the formula becomes \( W = Fd \cos \theta \), where \( \theta \) is the angle between the force vector and the displacement vector. If the angle is 0°, the cosine equals 1, and the entire force contributes to the work. If the angle is 90°, the cosine equals 0, and no work is done, even if a force is applied.

• When pushing or pulling at an angle, such as dragging a sled with a rope, only part of the force actually moves the sled forward. The vertical component of the force may reduce the normal force, but it does not contribute to forward motion.
• The cosine factor ensures that we only count the force component parallel to the displacement. This prevents overestimating work when forces are applied at an angle, which is a common mistake in AP problems.
• If a force is applied upward at an angle, the vertical component decreases the normal force, which can also reduce friction. Thus, angled forces often affect multiple aspects of motion, not just the net work done.
• In some cases, a force can do both positive and negative work simultaneously. For example, a person pulling a sled uphill at an angle does positive work through the forward component but increases frictional effects through the added vertical lift.
• AP students often confuse total force with useful force in angled problems. Always break the force into horizontal and vertical components, and remember that only the horizontal component (parallel to motion) contributes to work.

Example: A 50 N force is applied at a 30° angle above the horizontal to move a block 2 m across a floor. The work is:
\( W = Fd \cos \theta = 50 \cdot 2 \cdot \cos(30°) \approx 86.6 \, J \).

Mechanical Energy

Mechanical energy is the sum of an object’s kinetic and potential energy. It represents the total energy available for doing mechanical work, such as moving, lifting, or compressing a spring. In AP Physics 1, mechanical energy is a central focus because many problems involve analyzing how energy shifts between kinetic and potential forms. While total energy in the universe is always conserved, mechanical energy is only conserved if no nonconservative forces (like friction or air resistance) act on the system. Understanding when mechanical energy is conserved and when it isn’t is crucial for correctly solving AP exam problems.

\[ E_{\text{mech}} = KE + PE \]

• If only conservative forces (like gravity and springs) act, mechanical energy is conserved. In this case, energy may shift from potential to kinetic or vice versa, but the total remains constant. For example, a roller coaster converts gravitational potential energy into kinetic energy as it descends without losing any mechanical energy.
• When nonconservative forces (such as friction or air resistance) act, they remove mechanical energy from the system. This lost mechanical energy usually becomes thermal energy or sound. In AP Physics, this requires including the work done by nonconservative forces in calculations.
• Mechanical energy conservation is often the fastest way to solve problems because it avoids calculating acceleration or time. Instead, you can equate initial and final energies to find unknowns like speed or height.
• The choice of zero for potential energy does not affect the total mechanical energy change. What matters are the differences between energy states, not the absolute values. This allows flexibility in problem-solving as long as consistency is maintained.
• Common AP scenarios involving mechanical energy include objects sliding on ramps, pendulums, mass-spring systems, and projectiles (neglecting air resistance). Each involves analyzing how energy is stored, transferred, and sometimes dissipated.

Potential Energy

Potential energy (PE) is stored energy that an object possesses because of its position, configuration, or condition. The most common form in AP Physics 1 is gravitational potential energy, which depends on height above a reference level. Another important form is elastic potential energy, stored in stretched or compressed springs. Potential energy allows systems to store work that can later be released as kinetic energy, making it a cornerstone of energy conservation problems. The zero level of potential energy is arbitrary, but consistency is essential when solving problems.

Gravitational Potential Energy

\[ PE_g = mgh \]

• Gravitational potential energy depends on mass, the acceleration due to gravity \( g \), and the object’s height above a reference point. Increasing height increases potential energy linearly.
• The choice of reference point for height is arbitrary. You can set the ground as \( h = 0 \), or the lowest point of a roller coaster, as long as you remain consistent throughout the problem.
• Raising an object higher stores more potential energy, which can later convert into kinetic energy. For example, lifting a ball before dropping it gives it potential energy that is released as motion.
• Work done against gravity to lift an object is equal to the increase in its gravitational potential energy. This explains why lifting a book requires energy even if the book’s speed remains zero.
• Potential energy is always relative, not absolute. Saying an object has “zero” potential energy simply means it is at the chosen reference level, not that it has no energy at all.

Elastic Potential Energy

\[ PE_s = \frac{1}{2}kx^2 \]

• Elastic potential energy is stored when a spring (or other elastic material) is compressed or stretched from its equilibrium position. The formula involves the spring constant \( k \) and displacement \( x \).
• The energy increases with the square of displacement, meaning doubling the stretch quadruples the stored energy. This makes springs very sensitive to compression or stretching.
• The spring constant \( k \) represents stiffness. A stiffer spring requires more force to stretch a given distance and stores more energy for the same displacement.
• This type of potential energy is conservative: when the spring returns to its equilibrium length, all the stored energy can be released as kinetic energy if no nonconservative forces (like friction) interfere.
• Examples include archery bows, trampolines, and pogo sticks. AP problems often combine gravitational and elastic potential energy in conservation scenarios.

The Work-Energy Theorem

The Work-Energy Theorem establishes a direct connection between the net work done on an object and the change in its kinetic energy. It states that the total work performed by all forces on an object is equal to the change in the object’s kinetic energy:

\[ W_{\text{net}} = \Delta KE = KE_f - KE_i \]

• This theorem provides an alternative to directly using Newton’s Second Law. Instead of calculating forces and accelerations step by step, we can focus on energy changes to solve many problems more efficiently.
• Positive net work increases kinetic energy, making the object move faster, while negative net work decreases kinetic energy, slowing the object down. If no net work is done, the object’s speed remains constant.
• All forces acting on the object must be considered, including both conservative forces (like gravity) and nonconservative forces (like friction). Only the vector sum of their work equals the change in kinetic energy.
• The Work-Energy Theorem is especially powerful when analyzing motion over a distance, because it eliminates the need to compute acceleration and time explicitly. Instead, it relates force and displacement directly to speed changes.
• In AP Physics 1, this theorem is frequently tested in problems involving ramps, pulleys, or surfaces with friction, because it streamlines calculations and emphasizes the energy perspective over the force perspective.

Conservation of Energy

The Law of Conservation of Energy states that the total energy of an isolated system remains constant. In other words, energy cannot be created or destroyed; it can only change forms, such as from potential energy to kinetic energy. In AP Physics 1, we focus primarily on the conservation of mechanical energy, which includes kinetic and potential energy. When only conservative forces act on a system, the total mechanical energy remains constant. However, when nonconservative forces like friction or air resistance are present, mechanical energy decreases as thermal or other forms of energy increase.

\[ KE_i + PE_i = KE_f + PE_f \quad \text{(if only conservative forces act)} \]

\[ KE_i + PE_i + W_{\text{nc}} = KE_f + PE_f \quad \text{(if nonconservative forces act)} \]

• When conservative forces like gravity or spring forces act, the total mechanical energy remains constant. A falling ball converts potential energy to kinetic energy without any energy loss, provided air resistance is negligible.
• Nonconservative forces remove mechanical energy from the system. For example, friction converts some of the object’s energy into heat, reducing the sum of kinetic and potential energy but not violating total energy conservation.
• The choice of reference point for potential energy does not affect energy conservation. Even if you shift the zero level of potential energy, the differences in energy and the physical outcomes remain the same.
• Energy bar charts or “LOL” diagrams (List of forms, Organization, Location) are often useful tools in AP Physics 1 to visually represent how energy is stored and transferred throughout a problem.
• In AP problems, always identify whether nonconservative forces are present. If they are, include their work in the energy conservation equation; otherwise, you may incorrectly assume total mechanical energy remains constant.

Conservation of Energy with Nonconservative Forces

In real-world systems, forces like friction and air resistance often act on objects, making them nonconservative forces. Unlike conservative forces such as gravity or springs, nonconservative forces remove mechanical energy from the system by converting it into other forms such as heat, sound, or deformation. The modified conservation of energy principle accounts for this by including the work done by nonconservative forces. This ensures that while mechanical energy may not be conserved, total energy is still conserved — it simply changes into less useful forms. Recognizing when to include nonconservative work is critical for solving AP Physics 1 problems accurately.

\[ KE_i + PE_i + W_{\text{nc}} = KE_f + PE_f \]

• Nonconservative forces such as friction, air resistance, and applied pushes or pulls can add or remove energy from a system. For example, friction on a sliding block transforms some mechanical energy into heat, lowering the block’s speed compared to an ideal case.
• The term \( W_{\text{nc}} \) represents the work done by nonconservative forces. If this work is negative (like friction opposing motion), it reduces mechanical energy. If it is positive (like a person pushing), it increases mechanical energy.
• Even when mechanical energy decreases, the total energy of the universe remains constant. The “lost” energy doesn’t disappear; it has simply transformed into thermal, acoustic, or other non-mechanical forms.
• In AP Physics 1, it is crucial to check whether the problem specifies surfaces with friction, resistive forces, or external pushes. These always require adding a nonconservative term to the energy equation to avoid overestimating speed or height.
• Real-world examples include a skateboarder slowing down on a rough ramp due to friction, a car losing energy to air resistance while driving, or a ball bouncing lower after each bounce because energy is lost to sound and deformation.

Power

Power is the rate at which work is done or the rate at which energy is transferred. Unlike work, which measures the total energy transfer, power measures how quickly that transfer happens. The unit of power is the watt (W), where one watt equals one joule per second. Power can be calculated using average values or instantaneous values, depending on whether we want the overall rate of energy transfer or the rate at a specific moment in time. This makes power especially important in real-world applications such as engines, electrical systems, and athletics.

\[ P = \frac{W}{t} \quad \text{(average power)} \qquad P = Fv \quad \text{(instantaneous power, when force is parallel to velocity)} \]

• Average power is found by dividing the total work done by the total time it took to do that work. This is useful when we want to know overall performance, such as the average output of a car engine during a trip.
• Instantaneous power uses force and velocity at a specific instant. For example, a sprinter may produce high instantaneous power at the start of a race even though their average power over the whole race is lower.
• When force and velocity are not parallel, only the component of velocity in the direction of the force contributes to power. This ensures accuracy in cases where motion is angled or curving.
• Power is a scalar quantity, but its magnitude is critical in engineering. Machines are rated in horsepower or watts, and exceeding their power limit can cause failure or inefficiency.
• In AP Physics 1, problems often compare the power output of different systems or ask students to calculate how much faster work can be done with a more powerful engine or motor.

Efficiency

Efficiency measures how effectively a system converts input energy or power into useful output. Because some energy is always lost to non-useful forms such as heat, sound, or vibration, no real machine is 100% efficient. Efficiency is usually expressed as a percentage, comparing the useful output work or power to the total input work or power. This concept is important in AP Physics because it shows how energy conservation works in real-world systems that are not perfectly ideal. Understanding efficiency helps explain why larger forces or more energy are required in practice compared to theoretical calculations.

\[ \text{Efficiency} = \frac{W_{\text{out}}}{W_{\text{in}}} \times 100\% \quad \text{or} \quad \text{Efficiency} = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\% \]

• Efficiency accounts for the fact that not all input energy becomes useful work. For example, in a car engine, much of the chemical energy from fuel is lost as heat and sound, leaving only a fraction as useful kinetic energy.
• High efficiency means less energy is wasted. An efficiency of 90% indicates that only 10% of the input energy is lost to unwanted forms, while the rest becomes useful work or power.
• No real system can be 100% efficient because some energy will always dissipate as heat due to friction, resistance, or other nonconservative forces. This is a direct consequence of the Second Law of Thermodynamics.
• In AP Physics problems, efficiency often requires identifying the useful work done versus the total energy provided. This emphasizes the importance of carefully tracking where energy goes in a system.
• Everyday examples include light bulbs (where most input energy becomes light but some becomes heat), car engines, and household appliances. Comparing efficiency ratings helps consumers choose more energy-saving devices.

Example Problems for Unit 3: Work, Energy, and Power

Problem 1: Work Done at an Angle

A student pulls a 20 kg box 5.0 m across a horizontal floor using a rope that makes a 30° angle above the horizontal. The tension in the rope is 50 N, and friction is negligible. How much work does the student do on the box?

Solution:

Step 1: Use the work formula with an angle.
\( W = Fd \cos \theta \)
Step 2: Substitute values.
\( W = 50 \times 5.0 \times \cos(30°) \)
Step 3: Calculate.
\( W = 250 \times 0.866 \approx 216.5 \, \text{J} \)

Answer: The student does approximately \( 216.5 \, \text{J} \) of work.

Problem 2: Gravitational Potential Energy Conversion

A 2.0 kg ball is dropped from a height of 10 m. Neglecting air resistance, find the speed of the ball just before it hits the ground using conservation of energy.

Solution:

Step 1: Write the conservation of mechanical energy equation.
\( KE_i + PE_i = KE_f + PE_f \)
Step 2: At the start, \( KE_i = 0 \), \( PE_i = mgh \). At the bottom, \( PE_f = 0 \).
So, \( mgh = \tfrac{1}{2}mv^2 \).
Step 3: Solve for v.
\( v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 10} \approx 14.0 \, \text{m/s} \)

Answer: The ball’s speed is about \( 14.0 \, \text{m/s} \).

Problem 3: Nonconservative Forces (Friction)

A 5.0 kg block slides 8.0 m across a rough horizontal surface. The coefficient of kinetic friction is 0.25. If the block’s initial speed was 6.0 m/s, what is its final speed?

Solution:

Step 1: Find work done by friction.
\( F_k = \mu_k F_N = \mu_k mg = 0.25(5.0)(9.8) = 12.25 \, N \).
Work: \( W_{nc} = -F_k d = -12.25 \times 8.0 = -98 \, J \).

Step 2: Use modified energy conservation.
\( KE_i + W_{nc} = KE_f \).
\( \tfrac{1}{2}(5.0)(6.0^2) - 98 = \tfrac{1}{2}(5.0)v_f^2 \).

Step 3: Calculate.
Initial KE = 90 J. So, \( 90 - 98 = \tfrac{5}{2} v_f^2 \).
\( -8 = 2.5 v_f^2 \). Since result is negative, the block stops before 8 m.
This means friction is strong enough to bring the block to rest before the distance is completed.

Answer: The block comes to rest before traveling 8.0 m.

Problem 4: Power Output

A 70 kg sprinter runs up a 10 m tall hill in 8.0 s at constant speed. What is the average power output of the sprinter?

Solution:

Step 1: Calculate work done against gravity.
\( W = mgh = 70 \times 9.8 \times 10 = 6860 \, J \).

Step 2: Use average power formula.
\( P = \tfrac{W}{t} = \tfrac{6860}{8.0} \approx 857.5 \, W \).

Answer: The sprinter’s average power output is approximately \( 858 \, W \).

Problem 5: Efficiency

A motor receives 2000 W of electrical power but only produces 1600 W of useful mechanical power. Calculate its efficiency.

Solution:

Step 1: Use efficiency formula.
\( \text{Efficiency} = \tfrac{P_{\text{out}}}{P_{\text{in}}} \times 100\% \).

Step 2: Substitute values.
\( \text{Efficiency} = \tfrac{1600}{2000} \times 100\% = 80\% \).

Answer: The motor is 80% efficient.

Multi-Step Challenge Problems for Unit 3

Problem 1: Block on a Rough Incline with Spring

A 3.0 kg block slides down a rough incline that makes a 30° angle with the horizontal. The incline is 4.0 m long, and the coefficient of kinetic friction is 0.20. At the bottom, the block compresses a spring with spring constant \( k = 400 \, \text{N/m} \). Find: (a) the speed of the block at the bottom before hitting the spring, and (b) the maximum compression of the spring.

Solution:

Step 1: Find the net work done on the block as it moves down the incline.
Gravitational component: \( W_g = mg \sin \theta \times d \).
\( W_g = (3.0)(9.8)\sin(30°)(4.0) = 58.8 \, J \).
Frictional work: \( W_f = -\mu_k mg \cos \theta \times d \).
\( W_f = -(0.20)(3.0)(9.8)\cos(30°)(4.0) \approx -20.4 \, J \).

Step 2: Apply the Work-Energy Theorem to find speed at bottom.
\( KE_f - KE_i = W_g + W_f \).
Initial KE = 0.
\( \tfrac{1}{2}(3.0)v^2 = 58.8 - 20.4 = 38.4 \).
\( v = \sqrt{\tfrac{2(38.4)}{3.0}} \approx 5.06 \, \text{m/s} \).

Step 3: Use energy conservation to find spring compression.
\( KE = PE_s \).
\( \tfrac{1}{2}(3.0)(5.06^2) = \tfrac{1}{2}(400)x^2 \).
\( 38.4 = 200x^2 \).
\( x = \sqrt{\tfrac{38.4}{200}} \approx 0.44 \, \text{m} \).

Answer: (a) The block’s speed at the bottom is about \( 5.06 \, \text{m/s} \). (b) The spring compresses approximately \( 0.44 \, \text{m} \).

Problem 2: Power and Efficiency on an Incline

A 1500 kg car drives up a 100 m long incline at a constant speed of 15 m/s. The incline is at an angle of 10° with the horizontal, and the engine has an efficiency of 75%. (a) Find the minimum power output required from the engine. (b) Calculate the actual power input to the engine due to its efficiency.

Solution:

Step 1: Calculate the height gained by the car.
\( h = d \sin \theta = 100 \sin(10°) \approx 17.36 \, m \).

Step 2: Find the work done against gravity.
\( W = mgh = (1500)(9.8)(17.36) \approx 2.55 \times 10^5 \, J \).

Step 3: Calculate the time taken.
\( t = \tfrac{d}{v} = \tfrac{100}{15} \approx 6.67 \, s \).

Step 4: Find useful power output.
\( P_{\text{out}} = \tfrac{W}{t} = \tfrac{2.55 \times 10^5}{6.67} \approx 3.82 \times 10^4 \, W \).

Step 5: Account for efficiency to find input power.
\( \text{Efficiency} = \tfrac{P_{\text{out}}}{P_{\text{in}}} \).
\( P_{\text{in}} = \tfrac{P_{\text{out}}}{\text{Efficiency}} = \tfrac{3.82 \times 10^4}{0.75} \approx 5.09 \times 10^4 \, W \).

Answer: (a) The engine must provide at least \( 3.82 \times 10^4 \, W \) of useful power. (b) With 75% efficiency, the engine consumes about \( 5.09 \times 10^4 \, W \) of input power.

Common Misconceptions in Unit 3: Work, Energy, and Power

Work

• Misconception: If you push on an object and feel tired, you must be doing work.
  Correction: In physics, work requires displacement. If the object doesn’t move, no work is done, regardless of how much effort you feel. This distinction between physical and physiological effort is crucial on the AP exam.
• Misconception: Work is always positive if a force is applied.
  Correction: Work can be negative if the force opposes motion. For example, friction always does negative work when an object slides, reducing its kinetic energy.

Kinetic and Potential Energy

• Misconception: A heavier object always has more kinetic energy than a lighter one at the same speed.
  Correction: While mass affects kinetic energy, speed has a greater impact because it is squared in the formula. A small object moving very fast can have more kinetic energy than a larger, slower object.
• Misconception: The zero point for potential energy is fixed and absolute.
  Correction: Potential energy is relative. You can choose any reference level for \( h = 0 \), as long as you remain consistent throughout the problem.

Conservation of Energy

• Misconception: Mechanical energy is always conserved.
  Correction: Mechanical energy is conserved only when no nonconservative forces act. With friction or air resistance, mechanical energy decreases, though total energy is still conserved by transforming into heat or sound.
• Misconception: Energy is “lost” when friction or resistance acts.
  Correction: Energy is never destroyed. It is transformed into non-mechanical forms like heat or sound. The law of conservation of energy always holds.

Power and Efficiency

• Misconception: More powerful machines always use more energy.
  Correction: Power measures the rate of doing work, not the total work. A powerful machine can do the same work in less time, but energy use depends on the total work done, not just the power rating.
• Misconception: A machine with 100% efficiency is possible if built well enough.
  Correction: No real machine is 100% efficient. Some energy is always lost to heat, sound, or other non-useful forms, as dictated by the Second Law of Thermodynamics. Efficiency only approaches 100% in idealized models.