Understanding Conservation of Energy and Motion

The formula for kinetic energy is: $KE = \frac{1}{2} m \cdot v^2$

Where:

• $m$ is the mass (kg)
• $v$ is the velocity (m/s)

Exam Tip: When calculating kinetic energy, always remember to square the velocity before multiplying by the mass and the constant. A common mistake is multiplying the mass by the velocity and then squaring the whole result!

Total Mechanical Energy (TME)

When we look at an object moving through a system, we often want to know the "total" amount of energy available to it. This is known as Total Mechanical Energy (TME). TME is simply the sum of the potential energy and the kinetic energy at any given moment.

$TME = PE + KE$

In an ideal system where no energy is lost to friction or air resistance, the TME remains constant throughout the entire motion. This means that as an object falls, its PE decreases while its KE increases, but their sum—the TME—stays exactly the same.

Energy Transfers and the Law of Conservation

The Law of Conservation of Energy states that energy can neither be created nor destroyed, only changed from one form to another. This is most clearly seen in Energy Transfers.

Consider a cart rolling down a hill:

1. At the top of the hill, the cart has maximum PE and minimum KE.
2. As it descends, the height decreases (losing PE), but the speed increases (gaining KE). The PE is being transferred into KE.
3. At the bottom of the hill, the PE is at its minimum, and the KE is at its maximum.

This transfer is also visible in a pendulum. When you raise a pendulum to one side, you give it potential energy. As it swings through the bottom of its arc, that energy has been converted into kinetic energy.

Real-World Factors: Air Resistance

In textbook problems, we often assume a "frictionless" environment to make calculations simpler. However, in the real world, objects encounter air resistance.

Air resistance is a type of friction caused by an object moving through air molecules. This force converts some of the object's mechanical energy into thermal energy (heat), which is why a bouncing basketball will never bounce back to its original height. While the total energy of the universe is still conserved, the mechanical energy (the sum of PE and KE) decreases because some of it has "leaked" out into the environment as heat.

Worked Example: The Falling Rock

To master these concepts, you must practice converting between PE and KE using the conservation of energy.

Problem: A 1.5 kg rock is falling at 6 m/s and is currently 35 m off the ground. What is its velocity when it hits the ground? (Ignore air resistance).

Step 1: Calculate Total Mechanical Energy (TME) at the current position. First, find the current PE: $PE = m \cdot g \cdot h$ $PE = 1.5 \text{ kg} \cdot 9.8 \text{ m/s}^2 \cdot 35 \text{ m} = 514.5 \text{ J}$

Next, find the current KE: $KE = \frac{1}{2} m \cdot v^2$ $KE = 0.5 \cdot 1.5 \text{ kg} \cdot (6 \text{ m/s})^2$ $KE = 0.5 \cdot 1.5 \cdot 36 = 27 \text{ J}$

Now, find the TME: $TME = 514.5 \text{ J} + 27 \text{ J} = 541.5 \text{ J}$

Step 2: Use TME to find the final velocity. At the ground, the height is 0, so $PE = 0$. Therefore, all the TME must be converted into KE. $TME = KE_{\text{final}}$ $541.5 \text{ J} = \frac{1}{2} m \cdot v^2$ $541.5 = 0.5 \cdot 1.5 \cdot v^2$ $541.5 = 0.75 \cdot v^2$ $v^2 = \frac{541.5}{0.75} = 722$ $v = \sqrt{722} \approx 26.87 \text{ m/s}$

By using the principle that energy is conserved, we can solve for unknown speeds and heights without needing to know the time elapsed or the acceleration at every single point in the fall.