Law of Conservation of Energy - Ninth Grade Science
Across the four-and-a-half-billion-year history of planet Earth, every physical, chemical, and biological event—from the eruption of supervolcanoes and the collision of tectonic continents to the photosynthesis of oak leaves and the firing of human neurons—has obeyed an inviolable cosmic rule. Energy can change its costume, transforming from radiant sunlight into chemical bonds, from chemical bonds into mechanical work, and from mechanical work into heat; but the total quantity of energy in an isolated system remains strictly, perpetually constant. The Law of Conservation of Energy is arguably the most universal and unbreakable physical law ever discovered by science. In this chapter, you will investigate how energy transforms across conservative and non-conservative systems, track the mechanical interchange between kinetic and potential energy in free fall and pendulums, and analyze real-world energy dissipation.
Statement of the Law of Conservation of Energy
The Law of Conservation of Energy states that energy can neither be created nor destroyed; it can only be transformed from one form into another, or transferred from one system to another, such that the total energy of an isolated system remains completely invariant.
+--------------------------------------------------------------------------+
| Law of Conservation of Energy |
+--------------------------------------------------------------------------+
| In any isolated physical system: |
| Total Initial Energy = Total Final Energy |
| E_initial = E_final |
| Delta E_system = 0 |
+--------------------------------------------------------------------------+
Universal Energy Transformations
[ Radiant Solar Energy ]
|
v (Photosynthesis)
[ Chemical Energy in Plants / Fossil Fuels ]
|
v (Combustion in Power Plant)
[ Thermal Energy in High-Pressure Steam ]
|
v (Turbine Rotation)
[ Mechanical Kinetic Energy ]
|
v (Electromagnetic Generator)
[ Electrical Energy ]
|
+-------------------+-------------------+
| | |
v v v
[ Light Bulbs ] [ Electric Heaters ] [ Electric Motors ]
(Radiant Light) (Thermal Heat) (Kinetic Energy)
No engine, chemical reaction, biological organism, or perpetual-motion machine can produce energy out of nothing; every joule of output work requires an equal intake of input energy.
Conservation of Mechanical Energy
In mechanics, when an isolated system experiences only conservative forces (such as gravity or ideal spring tension), the sum of its kinetic energy ($KE$) and potential energy ($PE$) remains constant at every point in time. This sum is called total mechanical energy ($E_{\text{mech}}$):
E_{\text{mech}} = KE + PE = \text{Constant}
KE_1 + PE_1 = KE_2 + PE_2
\frac{1}{2} m v_1^2 + m g h_1 = \frac{1}{2} m v_2^2 + m g h_2
Whenever potential energy decreases, kinetic energy must increase by the exact same amount ($\Delta KE = -\Delta PE$), and vice versa.
Case Study 1: The Freely Falling Body
Consider a solid object of mass $m$ dropped from rest from an initial height $H$ above the ground, falling under gravity in the absence of air resistance.
Conservation of Energy in Free Fall
Point A (Release from Rest, Height H):
Velocity v_A = 0 ===> KE_A = 0
Height = H ===> PE_A = m * g * H
Total Energy E_A = 0 + m * g * H = m * g * H
Point B (Intermediate Height h):
Velocity v_B = sqrt(2 * g * (H - h)) ===> KE_B = m * g * (H - h)
Height = h ===> PE_B = m * g * h
Total Energy E_B = m*g*(H - h) + m*g*h = m * g * H
Point C (Instant Before Ground Impact, Height = 0):
Velocity v_C = sqrt(2 * g * H) ===> KE_C = (1/2)*m*(2gH) = m * g * H
Height = 0 ===> PE_C = 0
Total Energy E_C = m * g * H + 0 = m * g * H
Mathematical Proof of Constancy
At Point A, the object possesses purely gravitational potential energy ($E = mgH$). As it descends toward Point B at height $h$, it has fallen through a distance $s = H - h$. Using the third kinematic equation ($v^2 = u^2 + 2as$), its velocity is $v_B^2 = 0 + 2g(H - h)$. Its kinetic energy at Point B is:
KE_B = \frac{1}{2} m v_B^2 = \frac{1}{2} m [2g(H - h)] = m g (H - h)
Adding its potential energy at Point B ($PE_B = mgh$):
E_B = KE_B + PE_B = m g (H - h) + m g h = m g H - m g h + m g h = m g H
At Point C immediately before impact, height is zero ($PE_C = 0$), and its velocity is $v_C^2 = 2gH$. Its kinetic energy is $KE_C = \frac{1}{2} m (2gH) = mgH$. The total mechanical energy at Point A, Point B, and Point C is identically $mgH$, proving that mechanical energy is conserved throughout the fall.
Case Study 2: The Oscillating Simple Pendulum
A swinging simple pendulum provides an elegant visual demonstration of continuous, periodic energy transformation.
The Oscillating Pendulum
Extreme Position (A) Extreme Position (B)
o o
/ \ / \
/ \ / \
/ \ / \
v v v v
[Bob] [Bob]
v = 0 v = 0
PE = Maximum PE = Maximum
KE = 0 KE = 0
|
v
Mean Position (O)
[Bob]
v = Maximum
PE = Minimum
KE = Maximum
The Energy Cycle
- At the extreme release positions (A and B), the pendulum bob momentarily halts ($v = 0$), so its kinetic energy is zero ($KE = 0$). All of its mechanical energy exists as gravitational potential energy ($PE = mgh$).
- As the bob swings downward from the extreme position toward the central mean position (O), its height decreases, converting potential energy into kinetic energy.
- At the mean position (O), the bob reaches its lowest vertical point ($h = 0$), so potential energy is at its minimum. Here, velocity reaches its maximum, and all the system's energy exists as kinetic energy ($KE = \frac{1}{2}mv_{ ext{max}}^2$).
- Inertia carries the bob through the mean position, causing it to climb toward the opposite extreme (B), converting kinetic energy back into potential energy. In the absence of air friction, this oscillation would continue perpetually.
Non-Conservative Forces and Energy Dissipation
In the real world, mechanical energy is rarely conserved completely because macroscopic systems are exposed to non-conservative forces, primarily friction, air resistance, and material viscosity.
The Extended Energy Conservation Equation:
E_mech,initial + W_non-conservative = E_mech,final
(KE_1 + PE_1) - W_friction = (KE_2 + PE_2)
When a car skids to a halt, its kinetic energy appears to vanish. However, energy is not destroyed; the negative work done by braking friction converts the organized kinetic energy of the car into disorganized thermal energy, heating the brake rotors, tires, and pavement, and radiating acoustic sound waves into the surrounding air. When all thermal, acoustic, and radiant energy forms are accounted for, the total universal energy balance remains precisely conserved down to the last decimal place.
Chapter Exercises
Exercise 1: A roller coaster cart with a total mass of 500 kilograms is pulled to the top of a towering hill 45.0 meters above the ground and released from rest. Assuming friction and air resistance along the track are completely negligible, calculate the gravitational potential energy of the cart at the summit relative to ground level, and determine the velocity of the cart when it reaches the bottom of a track dip situated 15.0 meters above the ground. Take $g = 9.8 ext{ m/s}^2$.
Exercise 2: A daredevil on a skateboard rolls off a 3.2-meter-high ramp with an initial horizontal speed of 2.0 meters per second. Calculate the skateboarder's speed upon reaching the bottom of the ramp at ground level, assuming mechanical energy is fully conserved.
Exercise 3: Explain why a child bouncing on a trampoline will gradually bounce to lower and lower heights and eventually come to rest if they do not actively pump their legs. Identify the energy transformations taking place and explain whether the Law of Conservation of Energy is violated.
Solutions and Explanations
Solution 1: At the top of the 45.0-meter hill (Point 1), the cart is released from rest, so $v_1 = 0$ and $KE_1 = 0$. The potential energy is $PE_1 = mgh_1 = 500 ext{ kg} imes 9.8 ext{ m/s}^2 imes 45.0 ext{ m} = 220500$ joules (220.5 kJ). The total mechanical energy is $E = 220500$ joules. At the intermediate dip (Point 2) at height $h_2 = 15.0 ext{ m}$, the potential energy is $PE_2 = mgh_2 = 500 imes 9.8 imes 15.0 = 73500$ joules. By conservation of mechanical energy, $E_1 = E_2 \implies 220500 = KE_2 + 73500$, which gives $KE_2 = 220500 - 73500 = 147000$ joules. Since $KE_2 = \frac{1}{2} m v_2^2$, we have $\frac{1}{2}(500) v_2^2 = 147000$, which gives $250 v_2^2 = 147000$, resulting in $v_2^2 = 147000 / 250 = 588$. Taking the square root: $v_2 = \sqrt{588} \approx 24.25$ meters per second.
Solution 2: Use the conservation of mechanical energy: $KE_1 + PE_1 = KE_2 + PE_2$. At the top of the ramp, height $h_1 = 3.2 ext{ m}$ and initial velocity $u = 2.0 ext{ m/s}$. At the bottom, $h_2 = 0$ ($PE_2 = 0$). Write the energy equation: $\frac{1}{2} m u^2 + m g h_1 = \frac{1}{2} m v^2 + 0$. Mass $m$ cancels out of every term: $\frac{1}{2} u^2 + g h_1 = \frac{1}{2} v^2$. Multiply the entire equation by 2: $v^2 = u^2 + 2 g h_1$. Substitute known values: $v^2 = (2.0)^2 + 2(9.8)(3.2) = 4.0 + 62.72 = 66.72$. Taking the square root: $v = \sqrt{66.72} \approx 8.17$ meters per second. The skateboarder arrives at the bottom at approximately 8.17 m/s.
Solution 3: The child's bouncing height diminishes because the trampoline-child system is not isolated from non-conservative dissipative forces. During each bounce cycle, mechanical energy is lost to three primary sinks: aerodynamic air resistance opposing the child's body in flight, internal friction and viscoelastic hysteresis within the stretching synthetic trampoline mat and steel springs, and acoustic energy radiating away as sound waves with each bounce. These non-conservative mechanisms convert organized mechanical energy (kinetic and gravitational potential) into disorganized thermal energy, slightly warming the mat, springs, surrounding air, and the child's body. The Law of Conservation of Energy is not violated; total universal energy remains strictly constant, but mechanical energy has degraded into thermal energy.