Question

If energy cannot be created or destroyed, what happens to the energy of a ball as it rolls down a hill and rests at the bottoen?

Short answer

The energy of the ball transforms from potential energy to kinetic energy as it rolls down the hill and finally into thermal energy due to friction when it comes to rest. Energy is conserved throughout the process, it just changes form.

Step-by-step solution

Identifying the Initial Energy

When the ball is at the top of the hill it has maximum potential energy due to its height, and no kinetic energy since it's not moving.

Energy During the Roll

As it begins to roll down, the potential energy begins to transform into kinetic energy. Therefore, the potential energy decreases as the kinetic energy increases.

Final Energy State

When it reaches the bottom and comes to rest, all its energy has converted to a form of thermal energy due to friction between the ball and the surface of the hill. It is dissipated into the surroundings.

Energy Conservation

According to the law of conservation of energy, the initial energy (potential energy) of the ball should equal to the final energy (thermal energy due to friction). Hence, no energy is lost, it is merely converted from one form to another.

Key concepts

Potential Energy

Potential energy is the energy an object possesses because of its position or state. A classic example of potential energy is a ball positioned at the top of a hill. The ball has the potential to roll down due to gravity, which would then convert that stored energy into motion, or kinetic energy. The formula to calculate gravitational potential energy is \( PE = mgh \), where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height from the ground. In the exercise, the ball's energy at the top of the hill is all potential energy, which becomes lesser as the ball rolls down the hill.

Imagine holding a stretched spring – it also contains potential energy, specifically elastic potential energy, which can be released when the spring snaps back to its normal shape. Similarly, a charged battery has electric potential energy. The key to understanding potential energy is recognizing that it's a state of readiness to do work, a stored function that becomes actual upon action.

Kinetic Energy

Kinetic energy is the energy of motion. When the ball from our exercise begins to roll down the hill, its potential energy is transformed into kinetic energy. The faster the ball moves, the more kinetic energy it has. The formula to calculate this is \( KE = \frac{1}{2}mv^2 \), with \( m \) representing mass and \( v \) standing for the velocity of the object. As the ball accelerates down the hill, its velocity increases, hence increasing its kinetic energy.

Another way to envision kinetic energy is through activities we see every day. A flowing river, a speeding car, or even the wind – all contain kinetic energy. Their energy comes from their respective movements and speed. In daily life, kinetic energy is harnessed in numerous ways, like turbines transformed by river currents into electricity, an essential energy transformation that powers our world.

Thermal Energy

Thermal energy is the energy that comes from the temperature of an object, which is the energy possessed by its particles, such as atoms and molecules. When the ball in our exercise reaches the bottom of the hill, it doesn't just stop; it has gained thermal energy due to friction. Friction occurs when two surfaces rub against each other, like the ball rolling across the ground, and it converts the ball’s kinetic energy into thermal energy.

The warmer an object is, the more thermal energy it has. Think of it as when you rub your hands together on a cold day. The friction created between your palms turns the kinetic energy from the motion of rubbing into heat, warming your hands. In many real-life engineering systems, minimizing unnecessary thermal energy due to friction is essential to maintain efficiency and effectiveness of the machinery.

Energy Transformation

Energy transformation is the process of changing one form of energy to another. In our exercise, the ball undergoes a significant energy transformation from potential energy to kinetic energy and finally to thermal energy. This transformation demonstrates the law of conservation of energy, which states that energy in an isolated system can be neither created nor destroyed; only transformed.

In nature and technology, energy transformations are fundamental to all processes. A car engine, for example, transforms chemical energy from fuel into kinetic energy to move the car. In ecosystems, plants convert sunlight into chemical energy through photosynthesis, which then can be used by animals when they eat the plants. Understanding this principle allows us to predict how energy will flow and be used in different systems, which is invaluable when solving problems related to energy usage, conservation, and efficiency.