Question

Two objects isolated from the rest of the universe collide and stick together. Does the system's final kinetic energy depend on the frame of reference in which it is viewed? Does the system's change in kinetic energy depend on the frame in which it is viewed? Explain your answers.

Short answer

The final kinetic energy of the system does not depend on the frame of reference, due to the principle of energy conservation. However, the change in kinetic energy of the system does depend on the frame of reference.

Step-by-step solution

Understanding frames of reference

In physics, different observers can view the same event from different frames of reference. This means that the descriptions of physical processes can differ, depending on the viewpoints of observers. However, physical laws are said to be invariant, or constant, for all inertial frames of reference.

Apply laws of conservation

According to the law of conservation of energy, the total energy of an isolated system remains constant. Kinetic energy is part of this total energy. Therefore, the total final kinetic energy (i.e., the kinetic energy after the two objects have collided and stuck together) must be the same in all frames of reference to satisfy the principle of energy conservation.

Consider change in kinetic energy

The change in kinetic energy is the difference between the initial and the final kinetic energy. The initial kinetic energy is different in different frames of reference i.e. it depends on the observer’s velocity relative to the objects. Therefore, the change in kinetic energy of the system will differ for observers in different frames of reference.

Key concepts

Frames of Reference

Imagine you are in a car and you throw a ball up in the air. To you, the ball seems to just go up and then down, landing right back in your hand. But to someone standing on the sidewalk watching your car go by, the ball appears to follow a curved path. Both observations are correct, but they are from different frames of reference.
In physics, a frame of reference refers to the perspective from which an observer measures and observes phenomena. It's like a stage on which all physical events play out, and the position or motion of objects is described relative to this stage.
Importantly, while the perspectives can change, the laws of physics do not. For isolated systems, like the two colliding objects in the exercise, inertial frames of reference describe situations where the laws of motion hold true, maintaining consistent descriptions of phenomena.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. The formula to calculate kinetic energy (KE)is given by KE = \frac{1}{2}mv^2,where $m$ is the mass of the object and $v$ is its velocity.
When two objects collide and stick together, like in our problem, they form a single system. This process transforms the kinetic energy.

• Before collision: Each object has its own kinetic energy.
• After collision: The combined mass moves with a new velocity, resulting in a different kinetic energy.

The crucial point here is that the final kinetic energy is constant when measured in any inertial frame because of the conservation of energy. Any observer, whether moving or at rest, would find the same total kinetic energy after the collision, even if it looks different initially.

Invariance of Physical Laws

One of the fundamental principles in physics is that physical laws are invariant across all frames of reference. This means that no matter where or how fast you are moving, the laws governing the behavior of objects remain the same. This principle provides a stable foundation for our understanding of the universe.
Even though kinetic energy can appear different between frames due to relative motion, the conservation of energy principle holds universally. This means:

• The total energy in an isolated system doesn't change when viewed from any frame.
• The physical process, such as collisions, adhere to these invariant laws regardless of the observer's standpoint.

This invariance assures us that although observations can vary, the core physics stays consistent, offering explanations and predictions that work universally. It's like having a reliable rulebook that everyone follows, no matter where or how they're observing the game from.