Conservation of Mechanical Energy
Students use mathematical models of kinetic and gravitational potential energy to predict the motion of an object in a system and evaluate the effects of energy losses.
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Defining the System
Before analyzing energy, define the system: the object or group of objects being studied. Also identify the surroundings and the system boundary separating them. Energy can move across that boundary through work, heating, or other transfers. For a falling ball, a useful system is the ball and Earth together. Gravitational potential energy is then stored in the interaction between the ball and Earth, while kinetic energy belongs to the moving ball. If air is excluded from the system, air resistance transfers energy out of the system. For example, consider a 0.50-kilogram ball released from a balcony. A model that includes the ball and Earth but excludes the air can track gravitational potential energy changing into kinetic energy while treating air resistance as an external energy transfer. A clearly defined system determines which energy changes must appear in the model.
Kinetic and Potential Energy
Kinetic energy is the energy of motion and is modeled by K = one-half mv squared, where m is mass in kilograms and v is speed in meters per second. Gravitational potential energy near Earth's surface is modeled by Ug = mgh, where g is approximately 9.8 meters per second squared and h is height above a chosen reference level. Both energies are measured in joules. Because kinetic energy depends on speed squared, doubling speed produces four times as much kinetic energy. For example, a 2.0-kilogram cart moving at 3.0 meters per second has 9.0 joules of kinetic energy. If the same cart is 5.0 meters above the reference level, it has 98 joules of gravitational potential energy. The zero-height reference is a modeling choice, but height differences determine physical changes in potential energy.
The Mechanical Energy Equation
Mechanical energy is the sum of kinetic energy and potential energy: Emech = K + Ug. In an isolated system with no friction, air resistance, or other nonconservative interactions, total mechanical energy remains constant. The conservation equation is Ki + Ugi = Kf + Ugf. This equation does not mean that kinetic and potential energy stay constant separately; they transform into each other. Consider a 500-kilogram roller-coaster car released from rest and descending 10 meters. Its gravitational potential energy decreases by 49,000 joules, so its kinetic energy increases by 49,000 joules. Solving one-half mv squared = mgh gives v = the square root of 2gh, or about 14.0 meters per second. The mass cancels, showing that the predicted speed depends on the vertical drop rather than the car's mass when resistive forces are ignored.
Predicting Speed and Height
Energy equations can predict an object's speed at a known height or its height at a known speed. First choose initial and final states, set a zero-height reference, and write the energy terms present at each state. Then substitute known values and solve the resulting equation. For example, suppose a ball is launched straight upward from ground level at 20.0 meters per second. At its maximum height, its speed is zero. Ignoring air resistance, one-half mv squared = mgh, so h = v squared divided by 2g. The predicted maximum height is about 20.4 meters. At a height of 10.0 meters, conservation gives one-half m times 20.0 squared = one-half mv squared + mg times 10.0. Solving predicts a speed of about 14.3 meters per second, whether the ball is moving upward or downward at that height.
Energy Losses and Efficiency
When friction, drag, or deformation is present, mechanical energy is not conserved because some energy becomes thermal energy, sound, or other forms. Total energy is still conserved. A useful model is Ki + Ugi = Kf + Ugf + Edissipated. Suppose a 20-kilogram sled begins with 1,960 joules of gravitational potential energy but has only 1,500 joules of kinetic energy at the bottom of a hill. The model assigns 460 joules to dissipated energy. Mechanical efficiency equals useful mechanical-energy output divided by energy input, multiplied by 100 percent. The sled's efficiency is about 76.5 percent. A computational model can vary friction and calculate resulting speed, energy loss, and efficiency. Designers can then compare marginal benefits and costs: reducing another 50 joules of loss may slightly increase speed, but expensive materials or maintenance may make that improvement economically inefficient.
