Conservation of Energy
Why this matters
Remember the feeling at the top of a roller coaster? That moment of silence right before the plunge, where you're holding your breath, high above the ground. Then, whoosh! You trade all that height for incredible speed as you rush to the bottom. You didn't get a push from a giant engine mid-drop; the ride just converted one thing (height) into another (speed).
This trade-off is the heart of one of the most powerful ideas in all of physics: the conservation of energy. It's like a fundamental rule of the universe's accounting system. In this lesson, we'll learn how to track this energy, predict speeds and heights, and see why choosing your "system" is the most important decision you'll make.
Concept overview
flowchart TD
A[Start: Analyze a motion problem] --> B{Is work done by non-conservative forces like friction?};
B -->|No| C{Is work done by external forces?};
B -->|Yes| D[Use Work-Energy Theorem: <br> W_nc = ΔE_mech];
C -->|No| E[Mechanical Energy is Conserved! <br> E_initial = E_final];
C -->|Yes| F[Use Work-Energy Theorem: <br> W_ext = ΔE_mech];
E --> G[K_i + U_gi + U_si = K_f + U_gf + U_sf];
Core explanation
Before we can say energy is "conserved," we have to answer a critical question: conserved where? The first and most important step in any energy problem is to define your system. A system is just the collection of objects we've decided to pay attention to.
Choosing Your System Changes Everything
Let's think about that roller coaster cart.
Scenario 1: The System is "Just the Cart" If our system is only the cart, what's acting on it? The track pushes up (normal force), and the entire planet Earth pulls down on it (gravity). From the cart's perspective, gravity is an external force. As the cart falls, this external force does positive work, increasing the cart's kinetic energy. So, for a system of "just the cart," its energy is not constant; it changes because the environment (Earth) is doing work on it. A system with only one object can only have kinetic energy.
Scenario 2: The System is "The Cart + Earth"
Now, let's be smarter. Let's define our system as the cart and the Earth together. In this larger system, the gravitational pull between the cart and Earth is an internal force. It's part of the system's interaction. This interaction stores energy, which we call gravitational potential energy (Ug).
Because the force is now internal, it can't change the total energy of the Cart-Earth system. It can only convert energy that's already there. This is the magic choice! When the only forces doing work are internal, conservative forces (like gravity), the total mechanical energy is conserved.
The Big Equation: E = K + U
Mechanical energy (E) is the total energy of motion and position. It's the sum of kinetic energy and potential energy.
E = K + U
- Kinetic Energy (
K): The energy of motion, calculated asK = ½mv². - Potential Energy (
U): Stored energy. For now, we'll focus on gravitational potential energy,Ug = mgh.
The law of conservation of mechanical energy says that for an isolated system with no non-conservative forces (like friction), the initial total energy is equal to the final total energy.
E_initial = E_final
K_initial + U_initial = K_final + U_final
This is your golden ticket for solving a huge number of physics problems without dealing with forces and acceleration directly.
Back to the Roller Coaster
Let's trace the energy on a frictionless ride, just like in the Shrutam visual. Our system is the Cart + Earth.
- Point A (Top)The cart is at its maximum height (
h_max) and is momentarily at rest (v=0).K_A = ½m(0)² = 0U_gA = mgh_max- Total Energy:
E_A = 0 + mgh_max = mgh_max
- Point B (Bottom)The cart is at its lowest point. Let's define this as our
h=0level. It's moving at its fastest speed (v_max).K_B = ½mv_max²U_gB = mg(0) = 0- Total Energy:
E_B = ½mv_max² + 0 = ½mv_max²
Since energy is conserved, E_A = E_B.
mgh_max = ½mv_max²
- Point C (Mid-Hill): The cart is at some intermediate height (
h_C) and has some speed (v_C).K_C = ½mv_C²U_gC = mgh_C- Total Energy:
E_C = ½mv_C² + mgh_C
Again, energy is conserved, so E_A = E_C.
mgh_max = ½mv_C² + mgh_C
The total energy E is the same at every single point on the ride. It's like you have a $100 budget. At the top, it's all in your bank account (potential). At the bottom, you've spent it all on something you're using (kinetic). In the middle, you have some cash in your wallet and some in the bank, but the total is still $100.
What Happens When Energy Isn't Conserved?
These are non-conservative forces. They take mechanical energy out of the system and turn it into thermal energy (heat). The track gets a little warmer, the air gets a little warmer. The total mechanical energy (K+U) decreases.
In this case, the change in mechanical energy is equal to the work done by these non-conservative forces.
W_nc = ΔE = E_final - E_initial
If an external force adds energy (like a booster on the track), W_ext would be positive and the system's total energy would increase. The key is that energy is never truly "lost"—it's just transferred out of our system or converted into a form we're not tracking, like heat. But for AP Physics 1, when a problem says "frictionless" or "negligible air resistance," that's your cue to use the simple conservation equation.
Worked examples
Speed at the Bottom of the Hill
Problem: A 500 kg roller coaster cart starts from rest at the top of a frictionless hill that is 40 m high. How fast is it traveling when it reaches the bottom of the hill?
Solution Walkthrough:
- 1Define the SystemOur system is the cart + Earth. This allows us to use conservation of mechanical energy because gravity is an internal force.
- 2Identify Initial and Final States
- Initial (i): At the top of the hill.
h_i = 40 m,v_i = 0 m/s. - Final (f): At the bottom of the hill. We'll set our reference level here, so
h_f = 0 m. We want to findv_f.
- Initial (i): At the top of the hill.
- 3Set up the Conservation EquationSince the track is frictionless, mechanical energy is conserved.
E_i = E_fK_i + U_gi = K_f + U_gf - 4Substitute the Formulas and Values
½mv_i² + mgh_i = ½mv_f² + mgh_f½(500 kg)(0)² + (500 kg)(9.8 m/s²)(40 m) = ½(500 kg)v_f² + (500 kg)(9.8 m/s²)(0) - 5Simplify and Solve
0 + 196,000 J = ½(500 kg)v_f² + 0196,000 J = (250 kg)v_f²v_f² = 196,000 / 250 = 784v_f = √784 = 28 m/s
The Skateboarder in the Half-Pipe
Problem: Priya, a 60 kg skateboarder, drops into a frictionless half-pipe from one edge, starting from rest. The edge is 3.0 m higher than the bottom of the pipe. At the very bottom of the pipe, she is moving at her maximum speed. How high up the opposite side will she go?
Solution Walkthrough:
- 1Define the SystemPriya + Earth.
- 2Identify States
- Initial (i): At the starting edge.
h_i = 3.0 m,v_i = 0 m/s. - Final (f): At the highest point on the opposite side. Her speed there will be momentarily zero, so
v_f = 0 m/s. We need to findh_f.
- Initial (i): At the starting edge.
- 3Set up the Conservation EquationNo friction means we can use conservation.
K_i + U_gi = K_f + U_gf - 4Substitute and Simplify
½mv_i² + mgh_i = ½mv_f² + mgh_f0 + mgh_i = 0 + mgh_fmgh_i = mgh_fh_i = h_fh_f = 3.0 m
Try it yourself
Problem 1: The Pendulum Swing
A heavy 2.0 kg bob is attached to a 1.5 m long string, making a pendulum. You pull it back so the string is horizontal and release it from rest. What is the speed of the bob as it passes through the lowest point of its swing?
Problem 2: Spring Launch
A 0.5 kg block is placed against a horizontal spring that has been compressed by 0.2 m. The spring constant is k = 200 N/m. After the block is released, it slides across a frictionless surface. What is the block's speed after it leaves the spring?
Practice — 8 questions
In simple terms, conservation of energy means that in a closed system, total energy isn't created or destroyed, it just changes form—like converting the energy of height into the energy of speed.
- 3.4.A: Describe the energies present in a system.
- 3.4.B: Describe the behavior of a system using conservation of mechanical energy principles.
- 3.4.C: Describe how the selection of a system determines whether the energy of that system changes.
- 3.4.A.1
- A system composed of only a single object can only have kinetic energy.
- 3.4.A.2
- A system that contains objects that interact via conservative forces or that can change its shape reversibly may have both kinetic and potential energies.
- 3.4.B.1
- Mechanical energy is the sum of a system's kinetic and potential energies.
- 3.4.B.2
- Any change to a type of energy within a system must be balanced by an equivalent change of other types of energies within the system or by a transfer of energy between the system and its surroundings.
- 3.4.B.3
- A system may be selected so that the total energy of that system is constant.
- 3.4.B.4
- If the total energy of a system changes, that change will be equivalent to the energy transferred into or out of the system.
- 3.4.C.1
- Energy is conserved in all interactions.
- 3.4.C.2
- If the work done on a selected system is zero and there are no nonconservative interactions within the system, the total mechanical energy of the system is constant.
- 3.4.C.3
- If the work done on a selected system is nonzero, energy is transferred between the system and the environment.
flowchart TD
A[Start: Analyze a motion problem] --> B{Is work done by non-conservative forces like friction?};
B -->|No| C{Is work done by external forces?};
B -->|Yes| D[Use Work-Energy Theorem: <br> W_nc = ΔE_mech];
C -->|No| E[Mechanical Energy is Conserved! <br> E_initial = E_final];
C -->|Yes| F[Use Work-Energy Theorem: <br> W_ext = ΔE_mech];
E --> G[K_i + U_gi + U_si = K_f + U_gf + U_sf];
Read what Saavi narrates
Hey everyone, it's Saavi.
Remember the feeling at the top of a roller coaster? That quiet moment right before the big drop, where you're way up high, holding your breath. Then, whoosh! You trade all that height for incredible speed. The ride didn't use an engine to push you down; it just converted the energy of being high up into the energy of moving fast.
That's the big idea today: the conservation of energy. It's a fundamental rule that says in a closed system, energy can't be created or destroyed, it just changes form. The total amount, what we call mechanical energy, stays constant. It's just the sum of kinetic energy, the energy of motion, and potential energy, the stored energy of position.
Let's walk through an example. Imagine a 500-kilogram roller coaster cart starting from rest at the top of a 40-meter high frictionless hill. We want to find its speed at the very bottom.
First, we define our system as the cart and the Earth. This is key. Because we did that, we can say the initial energy equals the final energy.
Initially, at the top, the cart is at rest, so its kinetic energy is zero. But it has a lot of potential energy, which is mass times g times height. So, 500 kilograms times 9.8 times 40 meters, which is 196,000 Joules.
At the bottom, the height is zero, so the potential energy is zero. All of that initial energy has been converted into kinetic energy, which is one-half m v-squared.
So, we set them equal: 196,000 Joules equals one-half times 500 kilograms times v-squared. A little bit of algebra... we divide by 250, take the square root, and we find the final velocity is 28 meters per second. See? No forces, no kinematics, just a simple energy balance.
Now, a really common mistake I see every year is students trying to apply this to every single problem. If a problem mentions friction or air resistance, you can't do this. Friction removes mechanical energy from the system, turning it into heat. In that case, the final energy will be less than the initial. So always check if the problem is frictionless first!
This is a powerful tool. Once you get the hang of it, you'll find it's one of the easiest ways to solve problems. Keep practicing, and you've got this.
If you choose "just the object" as your system, its energy is *not* conserved because gravity is an external force doing work. You must include the Earth in your system to use potential energy.
Always start a problem by stating your system is "the object + Earth." This justifies using the `K+U = constant` equation.
Potential energy depends on your chosen zero level. While any level works, the easiest is almost always the lowest point in the object's motion. Sticking to one `h=0` is critical for consistency.
Before writing any equations, draw a quick sketch and label the `h=0` line. Use that reference for all your height measurements.
The word "lost" is dangerous in physics. In a conservative system, energy is never lost; it's only *transformed* from one form (like potential) to another (like kinetic).
Use the language of "conversion" or "transformation." The bar charts for the roller coaster are a great mental model: the total bar height never changes, just the size of the K and U sections.
This is a very common algebra slip-up under pressure. You'll get an answer that is way off.
When you write `K = ½mv²`, say the "squared" part out loud in your head. Double-check your algebra before you plug in numbers.
Friction is a non-conservative force that removes mechanical energy from the system (turning it into heat). The total mechanical energy will decrease.
If a problem mentions friction, a rough surface, or air resistance, you cannot use `K_i + U_i = K_f + U_f`. You must use the Work-Energy Theorem: `W_nc = E_f - E_i`.