Energy Conservation- Practice Problems and Solutions

What Energy Conservation Actually Means

Energy conservation isn't some abstract principle your teacher invented to torture you. It's a hard rule: energy cannot be created or destroyed — only converted from one form to another. That's it. Everything else in physics flows from that simple statement.

In mechanics problems, you're usually tracking the conversion between kinetic energy (KE = ½mv²) and gravitational potential energy (PE = mgh). Friction complicates things by converting mechanical energy into thermal energy, but the total energy of a closed system stays constant.

The Core Equation You Need

For most practice problems, you'll use this relationship:

KE₁ + PE₁ = KE₂ + PE₂

Or, if friction is involved:

KE₁ + PE₁ = KE₂ + PE₂ + W(friction)

Where W(friction) = f × d (force times distance).

Energy Types at a Glance

Practice Problem #1: The Falling Object

Problem: A 2 kg ball drops from rest at height h = 10 m. Find its speed just before hitting the ground. Ignore air resistance.

Solution:

Initial state: ball at rest (v = 0), height h = 10 m

Final state: just before impact (height = 0)

Apply conservation of energy:

Initial PE = Final KE

mgh = ½mv²

Solve for v:

v² = 2gh

v² = 2(9.8)(10)

v² = 196

v = 14 m/s

That's your answer. No complicated kinematics needed. One equation, done.

Practice Problem #2: The Roller Coaster

Problem: A cart starts at rest at height 25 m. It rolls down a frictionless track. What is its speed at the bottom?

Solution:

Same setup as before. All potential energy converts to kinetic energy.

mgh = ½mv²

v = √(2gh)

v = √(2 × 9.8 × 25)

v = √490

v = 22.1 m/s

The mass cancels out completely. Every object falling from 25 m hits the bottom at the same speed regardless of how heavy it is. That's physics being honest with you.

Practice Problem #3: With Friction

Problem: A 3 kg block slides down a rough incline from height 5 m. The work done by friction is 80 J. Find the speed at the bottom.

Solution:

Initial energy: PE = mgh = (3)(9.8)(5) = 147 J

Final energy includes KE minus the energy lost to friction:

PE₁ = KE₂ + W(friction)

147 = ½(3)v² + 80

½(3)v² = 67

1.5v² = 67

v² = 44.67

v = 6.68 m/s

Friction always steals energy. The block gets to the bottom slower than it would on a smooth surface.

Practice Problem #4: The Spring

Problem: A 0.5 kg mass attached to a spring (k = 200 N/m) is compressed 0.2 m and released. What speed does it have when it passes through the equilibrium position?

Solution:

Elastic PE converts to kinetic energy:

½kx² = ½mv²

(200)(0.2)² = (0.5)v²

8 = 0.5v²

v² = 16

v = 4 m/s

Common Mistakes Students Make

• Forgetting to square the velocity — This is the most common error. When you solve for v, you need v² first, then take the square root.
• Mixing up units — Mass in kg, height in meters, velocity in m/s. Everything must be consistent.
• Ignoring friction when it's present — Check the problem statement. If it says "rough surface" or gives a friction value, you must include energy loss.
• Using the wrong height reference — Pick a convenient zero point. Usually the lowest point in the problem works best.
• Solving with kinematics when energy is faster — If you see height and want velocity, energy conservation is almost always the faster route.

How to Solve Any Energy Conservation Problem

Step 1: Identify Your Initial and Final States

What is the object doing at the start? At the end? Write down what you know about velocity and position for each.

Step 2: Choose Your Zero Reference Point

For gravitational PE, pick where h = 0. The ground, the lowest point of a track, wherever makes math easiest.

Step 3: Write the Energy Conservation Equation

Sum of initial energies = sum of final energies. Include every relevant form: kinetic, gravitational PE, elastic PE, and work done by external forces.

Step 4: Plug In What You Know

Substitute values. Leave variables for what you don't know yet.

Step 5: Solve Algebraically

Isolate the unknown. Check that your answer has correct units.

When to Use Energy vs. Forces

Energy conservation works best when:

• You're given heights and need velocities
• The path is complicated (doesn't need to be straight)
• Friction is present and quantified
• Springs or elastic materials are involved

Use kinematics and Newton's laws when:

• You're asked about time
• Acceleration is non-uniform
• Forces are given and you need to find motion
• The problem specifically asks for acceleration

Quick Reference Formulas

Kinetic Energy: KE = ½mv²

Gravitational Potential Energy: PE = mgh

Elastic Potential Energy: PE = ½kx²

Work-Energy Theorem: W = ΔKE

Conservation: KE₁ + PE₁ + W(ext) = KE₂ + PE₂

That's everything you need. Practice the four problems above until you can solve them without looking at the solutions. Energy conservation problems become automatic once you recognize the pattern.