Question

For a totally elastic collision between two objects, which of the following statements is (are) true? a) The total mechanical energy is conserved. b) The total kinetic energy is conserved. c) The total momentum is conserved. d) The momentum of each object is conserved. e) The kinetic energy of each object is conserved.

Short answer

a) The total mechanical energy is conserved. b) The total kinetic energy is conserved. c) The total momentum is conserved. d) The momentum of each object is conserved. e) The kinetic energy of each object is conserved. Answer: a), b), and c) are true, while d) and e) are false.

Step-by-step solution

Statement a: The total mechanical energy is conserved.

During a totally elastic collision, both kinetic and potential energy can change, but the total mechanical energy remains constant. Therefore, statement a is true.

Statement b: The total kinetic energy is conserved.

In a totally elastic collision, the total kinetic energy of the system (both objects) remains constant. After the collision, the individual kinetic energies may change, but the collective kinetic energy is conserved, making statement b true.

Statement c: The total momentum is conserved.

According to the law of conservation of momentum, the total momentum of a closed system is conserved before and after a collision. Since a totally elastic collision is an isolated event, this principle holds true for this case as well. Thus, statement c is true.

Statement d: The momentum of each object is conserved.

Even though the total momentum of the system is conserved, the individual momentum of each object may change as a result of the collision. Therefore, statement d is false.

Statement e: The kinetic energy of each object is conserved.

In an elastic collision, the total kinetic energy of the system is conserved, but not necessarily the kinetic energy of each object. After the collision, the individual kinetic energies may change. So, statement e is false. In summary, statements a), b), and c) are true, while statements d) and e) are false.

Key concepts

Conservation of Mechanical Energy

In the realm of physics, the principle of conservation of mechanical energy is fundamental, especially when examining elastic collisions. Mechanical energy consists of two major components: kinetic energy, which is the energy of motion, and potential energy, the energy stored by an object's position in a force field (typically a gravitational field). During an elastic collision, like the one described in our exercise, mechanical energy is conserved. This means the sum of kinetic and potential energy remains unchanged. For instance, two billiard balls colliding on a table conserve their mechanical energy because they only transfer kinetic energy to each other, while their sum total remains constant.

Conservation of Kinetic Energy

In the context of collisions, the conservation of kinetic energy is a specific case of the conservation of mechanical energy. Kinetic energy is the energy an object possesses due to its motion. In an elastic collision, which is perfectly efficient, with no energy lost to sound, heat, or deformation, the total kinetic energy before and after the crash is the same. However, it's essential to note that while the total kinetic energy for the whole system remains constant, the kinetic energy of each object can change. That's why, in our textbook exercise, although the total kinetic energy remains consistent after the collision, individual energies can vary; they essentially 'swap' energies depending on their masses and velocities.

Conservation of Momentum

The conservation of momentum is another crucial principle relevant to elastic collisions. Momentum, a measure of an object's motion that depends on the object's mass and velocity, is always conserved in a closed system. This physical law tells us that the total momentum of a system before the collision is equal to the total momentum after the collision.

A vivid illustration of conservation of momentum can be seen in Newton's cradle, a device with a series of swinging spheres. When one ball strikes the next, there is a transfer of momentum through the line of spheres, resulting in the ball at the opposite end swinging away. In our textbook problem, even though the momentum can switch between the colliding objects, the overall momentum of the system stays the same, in alignment with the law of conservation of momentum.

Physics of Collisions

Delving into the physics of collisions, we must distinguish between elastic and inelastic collisions. Elastic collisions are those in which both momentum and kinetic energy are conserved. This type of collision is often idealized because, in the real world, some energy transformation usually occurs, for example, into heat or sound. Inelastic collisions, on the other hand, do not conserve kinetic energy even though momentum still remains constant.

Understanding the types of collisions is crucial for solving problems related to crashes, explosions, and even quantum particle interactions. By identifying the nature of a collision, students can better predict the post-collision outcomes concerning velocity, energy, and momentum distribution.

Closed System Dynamics

When we refer to closed system dynamics, we consider a collection of objects where no external forces act on the system other than those that initiate the event. In the scenario of an elastic collision, we assume that no external forces such as friction, air resistance, or other environmental interactions significantly affect the system's energy or momentum during the short time frame of the collision.

A helpful way to visualize a closed system is to imagine two ice hockey pucks sliding and colliding on a very smooth ice surface with negligible friction. As a result of this 'closed' environment, the system conserves its total momentum and mechanical energy throughout the collision process.