Question

What kind of energy does ball A possess initially when at rest at the top of the hill? What kind of energies are involved as ball A moves down the hill? What kind of energy does ball A possess when it reaches the bottom of the hill and stops moving after hitting ball B? Where did the energy gained by ball B, allowing it to move up the hill, come from?

Short answer

Initially, ball A possesses gravitational potential energy, \( U_g = mgh \), when at rest at the top of the hill. As it moves down the hill, its gravitational potential energy decreases, and it gains kinetic energy, \( K = \frac{1}{2}mv^2 \). When it stops after hitting ball B at the bottom, the kinetic energy has been transferred to ball B, allowing it to move up the hill. This follows the conservation of energy principle, where the energy loss by ball A is equal to the energy gained by ball B.

Step-by-step solution

Identify the type of energy when ball A is at rest at the top of the hill

When ball A is at rest at the top of the hill, it possesses gravitational potential energy. It is given by the equation: \( U_g = mgh \) where \( m \) is the mass of ball A, \( g \) is the acceleration due to gravity, and \( h \) is the height of the hill.

Identify the types of energies as ball A moves down the hill

As ball A moves down the hill, its height decreases, meaning its gravitational potential energy decreases. At the same time, the ball gains kinetic energy due to its motion. Kinetic energy is given by the equation: \( K = \frac{1}{2}mv^2 \) where \( m \) is the mass of ball A and \( v \) is its velocity.

Identify the type of energy when ball A stops moving after hitting ball B

When ball A reaches the bottom of the hill and stops moving after hitting ball B, its gravitational potential energy has been completely converted into kinetic energy. However, since ball A stops after the collision, its kinetic energy has been transferred to ball B.

Explain where the energy gained by ball B comes from

The energy gained by ball B, allowing it to move up the hill, comes from the kinetic energy that was transferred to it during the collision. This transfer of energy follows the conservation of energy principle, which states that the total energy in a closed system remains constant. The energy loss by ball A is equal to the energy gained by ball B.

Key concepts

Gravitational Potential Energy

Imagine you are at the top of a slide, the anticipation building as you prepare to glide down. The position you're in provides you with a type of energy known as gravitational potential energy (GPE). This energy is a result of your height above the ground and the pull of Earth's gravity. Simply put, the higher you are, the more GPE you possess.

In physics, we calculate GPE using the formula: \( U_g = mgh \), where \( m \) stands for mass, \( g \) is the acceleration due to gravity, typically around \( 9.8 \) meters per second squared on Earth, and \( h \) represents the height above the ground. When an object is at rest at a certain height, just like a ball perched atop a hill, all its energy is gravitational potential energy.

Kinetic Energy

The thrill begins as you start descending the slide, and you're moving! This motion signifies that you now have kinetic energy (KE), the energy of an object in motion. Kinetic energy depends on two factors: the mass of the moving object and how fast it's going. For an object sliding down a hill or a ball rolling down a slope, kinetic energy increases as the object speeds up, which is because the gravitational potential energy is being converted into kinetic energy.

To understand how much kinetic energy an object has, we use the formula:\( K = \frac{1}{2}mv^2 \), with \( m \) as the mass and \( v \) as the velocity. As you reach the base of the slide or when the ball reaches the bottom of the hill, it has its maximum kinetic energy right before it starts to slow down or stop.

Energy Transfer

Let's say that at the bottom of the slide you bump into a friend, causing them to start moving. This illustrates energy transfer, which is a central theme in the conservation of energy principle. During a collision, energy doesn't disappear; it's simply passed from one object to another. In our example, the kinetic energy you had at the bottom of the slide was transferred to your friend, making them move.

Applying this to our ball scenario, when ball A hits ball B at the bottom of the hill, the kinetic energy of ball A is transferred to ball B. This energy allows ball B to move up the hill. It's important to note that some energy might be lost to friction or sound during the collision, but overall, the total energy within the system (ball A and B together) remains constant, just like your energy transferred to your friend. The principle of the conservation of energy ensures that the total energy before and after the transfer is the same.