The work-energy theorem connects forces and motion by showing how net work changes an object's kinetic energy. This cheat sheet helps students solve common physics problems involving pushes, pulls, friction, gravity, and changing speed. Worked-example style thinking is useful because many problems can be solved without finding acceleration or time first.
The central rule is . Work from a constant force is , while kinetic energy is . Conservative forces can also be handled with energy conservation, using and .
Key Facts
- The work-energy theorem is .
- Work done by a constant force is , where is the angle between the force and displacement.
- Kinetic energy is , so speed affects kinetic energy by the square of .
- Positive net work increases speed because means .
- Negative net work decreases speed because means .
- Gravitational potential energy near Earth's surface is , where is height relative to a chosen zero level.
- Friction usually does negative work given by when it acts opposite the motion.
- With nonconservative work, energy can be tracked using .
Vocabulary
- Work
- Work is energy transferred by a force acting through a displacement, calculated for a constant force by .
- Net Work
- Net work is the total work done by all forces on an object, written as .
- Kinetic Energy
- Kinetic energy is the energy of motion, calculated by .
- Work-Energy Theorem
- The work-energy theorem states that the net work on an object equals its change in kinetic energy, .
- Conservative Force
- A conservative force stores and returns energy, so its work can be represented by a change in potential energy.
- Nonconservative Force
- A nonconservative force, such as friction, changes mechanical energy by converting some energy to thermal energy or other forms.
Common Mistakes to Avoid
- Using force instead of net force is wrong because the theorem uses total work from all forces, so calculate before setting it equal to .
- Forgetting the angle in is wrong because only the component of force parallel to displacement does work.
- Treating friction work as positive is wrong when friction opposes motion, because its work is usually .
- Canceling mass automatically is wrong because mass cancels in some gravity-only problems but not when applied forces or friction terms depend differently on .
- Using instead of in kinetic energy is wrong because kinetic energy depends on speed squared, .
Practice Questions
- 1 A box starts from rest and has of net work done on it. What is its final speed?
- 2 A car slows from to . What net work was done on the car?
- 3 A student pushes a box with over at an angle of above the horizontal. How much work does the push do?
- 4 If two objects have the same speed but different masses, which has more kinetic energy, and how does the work-energy theorem explain your answer?
Understanding Work-Energy Theorem Worked Examples
A worked example becomes easier when you first choose the object or system being studied. Draw that object by itself, then list every force that has a component along its motion. A force at right angles to the motion does no work, even when the force is large.
For example, the normal force on a box sliding across a level floor usually does no work because it points upward while the box moves sideways. This helps separate forces that change speed from forces that only change direction or provide support.
Signs are often the hardest part. Do not assign a sign from the name of a force. Decide it from the force direction compared with the displacement.
Gravity does negative work on a ball moving upward, yet it does positive work on that same ball while it falls. Friction removes mechanical energy from the moving object because its force points opposite the displacement.
An applied push can be negative work if someone pulls backward while an object keeps moving forward. Write the direction of motion before deciding each work sign.
Energy methods are especially useful when a problem gives distances, heights, and speeds but says nothing about time. A skateboarder rolling down a ramp loses gravitational potential energy. That energy can become kinetic energy, making the skateboarder faster.
If the ramp is rough, some energy is transferred by friction into thermal energy in the board, wheels, ramp, and air. This is why a real skateboarder may reach a lower speed than a calculation without friction predicts. The same reasoning appears in braking cars, roller coasters, elevators, falling tools, and objects pulled across floors.
Check units at every step. Work and every form of energy use joules. A joule is the same unit whether it comes from a push, a change in height, or a change in speed.
Keep mass in kilograms, distance in metres, force in newtons, and speed in metres per second. Then test whether the final result makes physical sense. A negative change in kinetic energy cannot produce a larger final speed.
A calculated final kinetic energy below zero signals that the object stops before completing the stated motion, or that an assumption needs review. In multi-force problems, find the contribution from each force separately before combining them. This simple habit prevents missing friction, using the wrong distance, or counting gravity twice.