Question

Suppose a ball is sitting at the top of a hill. At this point, the ball has potential energy. The ball rolls down the hill, so the potential energy is converted into kinetic energy. When the ball reaches the bottom of the hill, it goes halfway up the hill on the other side and stops. If energy is supposed to be conserved, then why doesn't the ball go up the other hill to the same level as it started from?

Short answer

Friction and air resistance cause energy loss, preventing the ball from reaching the same height.

Step-by-step solution

Understand Energy Conservation

Energy conservation means that the total energy in a closed system remains constant. This means that the potential energy the ball has at the top of the hill should convert entirely into kinetic energy and vice versa, theoretically allowing the ball to reach the same height on the opposite hill.

Identify Energy Losses

In the real world, there are often energy losses due to factors such as friction and air resistance. These factors convert some of the ball's mechanical energy into other forms, such as heat, which is not recoverable.

Apply Energy Losses to the Scenario

As the ball rolls down the hill, some of its potential energy is converted into kinetic energy, but some is also lost to friction and air resistance. This means that when the ball reaches the bottom, it has less kinetic energy than it would in a frictionless scenario.

Determine the Ball's Height on the Opposite Hill

Because some energy has been lost, the ball cannot convert 100% of its remaining kinetic energy back into potential energy. As a result, the ball will not reach the same height on the opposite hill and will only go halfway when it stops.

Key concepts

potential energy

Potential energy is the energy an object has due to its position or condition. For example, when a ball sits at the top of a hill, it has gravitational potential energy. This is because it’s positioned high above the ground. The higher the object, the more potential energy it has. The formula for gravitational potential energy is given by:\(PE = mgh\)where:

• \(m\) is the mass of the object (in kilograms)
• \(g\) is the acceleration due to gravity (9.8 m/s²)
• \(h\) is the height above the ground (in meters)

When the ball is at the top of the hill, it has maximum potential energy due to its height.

kinetic energy

Kinetic energy is the energy an object has because of its motion. When our ball rolls down the hill, it gains speed and thus gains kinetic energy. The more speed it gains, the more kinetic energy it has. The formula for kinetic energy is:\(KE = \frac{1}{2} mv^2\)where:

• \(m\) is the mass of the object (in kilograms)
• \(v\) is the velocity of the object (in meters per second)

As the ball rolls down, potential energy is transformed into kinetic energy. By the time it reaches the bottom of the hill, most of its potential energy should have converted into kinetic energy, assuming no energy loss.

friction

Friction is the force that opposes motion between two surfaces that are in contact. When the ball rolls down the hill, friction between the ball and the ground is at work. This force resists the ball’s motion and converts some of the mechanical energy into heat:

• Mechanical energy -> Heat energy

This energy conversion is what we call energy loss due to friction. Because energy is being converted to heat, less energy remains for the ball to use to climb the other side of the hill.

air resistance

Air resistance, or drag, is the force that opposes an object’s motion through the air. Just like friction, air resistance reduces the ball's speed by converting kinetic energy into other forms of energy, like heat:

• Mechanical energy -> Heat energy

Even though air resistance might seem negligible, it can have a noticeable effect when calculated over distance. The ball loses some of its mechanical energy to air resistance as it travels down and up the hills. Thus, it won’t have enough energy left to reach the original height.

energy loss

Energy loss occurs when energy is converted from one form into another, which is not useful for the ball's motion. In our ball's journey down the hill and back up, energy is lost mainly due to:

• Friction (between the ball and the ground)
• Air resistance (drag)

Both of these forces convert some of the ball’s mechanical energy into heat, which cannot be used to make the ball climb back up the hill. That’s why the ball doesn’t reach the starting height on the opposite side.

mechanical energy

Mechanical energy is the sum of potential and kinetic energy in an object. For our ball example, the total mechanical energy is initially all potential energy at the hill’s top. As the ball rolls down, this potential energy gets converted to kinetic energy. The total mechanical energy (ignoring losses) remains constant:\(ME = PE + KE\)However, in the real world, some of this mechanical energy is transformed into heat due to friction and air resistance. This transformation results in less mechanical energy available for the ball to climb the other hill. Therefore, total mechanical energy decreases as the ball moves, leading to its eventual stop halfway up the other hill.