4cConservation of Energy
Discover how total mechanical energy is conserved, how energy transforms between kinetic and potential forms, and what friction does to a system.
Conservation of energy is one of the most powerful tools in all of physics — it lets you solve complex motion problems without needing to know every force along the way.
Lesson Overview
The law of conservation of energy states that energy cannot be created or destroyed — it can only be transformed from one form to another. Total mechanical energy is the sum of kinetic energy (KE) and potential energy (PE). In ideal systems with no friction, total mechanical energy remains constant. When friction is present, mechanical energy is converted to thermal energy, but total energy is still conserved.
Key Concepts
Law of Conservation of Energy
Energy cannot be created or destroyed; only transformed
Total Mechanical Energy
E = KE + PE = ½mv² + mgh
Energy Transformation
Conversion between KE, PE, thermal, chemical, etc.
Friction & Thermal Energy
Friction converts mechanical energy to heat (thermal energy)
Conservation Equation
KE₁ + PE₁ = KE₂ + PE₂ (no friction)
Work-Energy Theorem
Net work done on an object equals its change in KE
A 2 kg ball is dropped from rest at a height of 5 m. What is its speed just before it hits the ground? (g = 10 m/s²)
A 0.5 kg toy car rolls down a frictionless ramp from a height of 0.8 m. What is its kinetic energy at the bottom?
A pendulum bob has 12 J of potential energy at its highest point. What is its kinetic energy at the lowest point (no friction)?
A 3 kg object slides down a ramp of height 4 m. Due to friction, it arrives at the bottom with only 80 J of kinetic energy. How much energy was lost to friction? (g = 10 m/s²)
A 1 kg ball is launched upward with a speed of 20 m/s. What maximum height does it reach? (g = 10 m/s²)
A 4 kg rock is held at a height of 3 m. It is released from rest. Find its speed just before impact. (g = 10 m/s²)
Hint: Set PE at the top equal to KE at the bottom, then solve for v.
A skier starts from rest at the top of a 15 m hill. Assuming no friction, what is the skier's speed at the bottom?
Hint: Use mgh = ½mv². Notice the mass cancels out.
A spring stores 50 J of elastic potential energy. It launches a 0.25 kg ball. What is the ball's speed when it leaves the spring?
Hint: Set elastic PE equal to KE: 50 = ½mv². Solve for v.
A 2 kg block slides down a 5 m high ramp and reaches the bottom with 60 J of KE. How much energy was converted to thermal energy by friction?
Hint: Calculate the initial PE, then subtract the final KE.
At what height does a 1 kg ball have equal kinetic and potential energy if its total mechanical energy is 40 J?
Hint: If KE = PE, then each equals half the total energy. Use PE = mgh to find h.
Key Vocabulary
Conservation of Energy
The principle that the total energy of an isolated system remains constant; energy is neither created nor destroyed.
Example: A falling ball converts PE to KE, but the total mechanical energy stays the same (ignoring air resistance).
Mechanical Energy
The sum of an object's kinetic energy and potential energy: E = KE + PE.
Example: A roller coaster at the top of a hill has high PE and low KE; at the bottom it has low PE and high KE, but the same total mechanical energy.
Energy Transformation
The conversion of energy from one form to another, such as from potential to kinetic or from mechanical to thermal.
Example: Friction transforms the mechanical energy of a sliding box into thermal energy (heat).
Thermal Energy
The internal energy of an object due to the random motion of its particles; often produced when friction acts on an object.
Example: Rubbing your hands together converts mechanical energy into thermal energy, warming your hands.
Interactive Practice — 5 Questions
A ball rolls off a frictionless table. Which statement about its total mechanical energy is correct?
A 2 kg object falls from rest through a height of 10 m (g = 10 m/s²). What is its kinetic energy just before hitting the ground?
When friction acts on a sliding object, what happens to the "lost" mechanical energy?
At the lowest point of a pendulum swing (no friction), which statement is true?
Which equation correctly represents conservation of mechanical energy (no friction)?
Independent Practice
A 5 kg object is dropped from a height of 8 m. Calculate its speed just before it hits the ground. (g = 10 m/s²)
A 0.3 kg ball is thrown upward with an initial speed of 15 m/s. Find the maximum height it reaches. (g = 10 m/s²)
A 10 kg cart rolls down a 6 m high frictionless hill. At the bottom, it has 480 J of KE. How much energy was lost to friction?
Explain in your own words why a pendulum gradually slows down and stops in a real (non-ideal) situation.
★ A roller coaster car (mass 500 kg) starts from rest at a height of 40 m. At a later point on the track it is at a height of 10 m. If 15,000 J of energy has been lost to friction, find the speed of the car at the 10 m point. (g = 10 m/s²)
ChallengeCommon Mistakes
Assuming that when energy is "lost" to friction it simply disappears.
Energy is never destroyed — it converts to thermal energy. Total energy (including thermal) is always conserved.
Forgetting to set a reference height for gravitational PE, leading to sign errors.
Always define your reference level (usually the lowest point) and measure all heights from that level consistently.
Cancelling mass before checking whether it actually cancels in the equation.
Mass cancels in frictionless conservation problems (mgh = ½mv²), but NOT when friction is involved and energy lost is given in joules.
Math Tips
For frictionless problems: set PE_top = KE_bottom → mgh = ½mv² → v = √(2gh). Mass always cancels here.
For problems with friction: KE_final = PE_initial − W_friction. Calculate initial PE first, then subtract energy lost.