Question

If you jump off a table onto the floor, is your mechanical energy conserved? If not, where does it go? b) A car moving down the road smashes into a tree. Is the mechanical energy of the car conserved? If not, where does it go?

Short answer

If not, what happens to the lost energy? Answer: Mechanical energy is not conserved in both scenarios due to the presence of non-conservative forces like air resistance and impact force from the tree in the car crash scenario. The lost energy is primarily converted into thermal energy (heat) and sound energy. In the car crash scenario, some energy is also used to deform the car and is absorbed by the tree, causing damage.

Step-by-step solution

Determine forces involved

For the person jumping off the table, the main forces involved are gravity and air resistance (assuming there is no friction between the person and the table).

Identify conservative and non-conservative forces

Gravity is a conservative force, meaning its work is path-independent, and it does not contribute to the loss of mechanical energy. However, air resistance is a non-conservative force, as work done by it depends on the path taken.

Analyze energy conservation

Since there is at least one non-conservative force (air resistance) acting on the jumper, the mechanical energy will not be conserved, and some energy will be converted into other forms.

Determine where energy goes

The energy lost due to air resistance will be primarily converted into thermal energy (heat), which dissipates into the surrounding air and the person's body. #Problem 2: Car crashing into a tree#

Determine forces involved

For the car crashing into a tree, the main forces involved are gravity, air resistance, and the force exerted by the tree on the car during impact (which can be regarded as a type of external force).

Identify conservative and non-conservative forces

Again, gravity is a conservative force, while air resistance is a non-conservative force. The impact force from the tree is also a non-conservative force as the work done by it depends on the path taken.

Analyze energy conservation

Since there are non-conservative forces acting on the car (air resistance and the force from the tree), the mechanical energy will not be conserved, and some energy will be converted into other forms.

Determine where energy goes

The lost energy is primarily converted into thermal energy (heat) and sound energy. Some energy is also used to deform the car during the collision, and absorbed by the tree, causing damage to the tree.

Key concepts

Conservative Forces

Understanding conservative forces is crucial when analyzing a system's mechanical energy. Conservative forces, such as gravity, are forces whose work done on an object is independent of the path taken. The unique characteristic of these forces is that they can store mechanical energy into potential energy forms. When you jump off a table, the force of gravity does work on you, and your potential energy due to height is converted into kinetic energy as you accelerate towards the ground. The total mechanical energy (potential plus kinetic) would be conserved in a system with only conservative forces acting upon it. In our exercise, however, this is not the case as non-conservative forces also play a role.

Non-Conservative Forces

Conversely, non-conservative forces result in energy transformations that cannot be fully recovered. Their hallmark is their path dependency—different paths will amount to different amounts of work done by non-conservative forces. For example, air resistance is a non-conservative force experienced when falling. It dissipates part of your mechanical energy into the air as thermal energy, hence why a falling object does not bounce back to its original height. Similarly, when a car crashes into a tree, the forces during impact are non-conservative; energy is largely lost to heat and the irreversible deformation of the car's structure.

Energy Conversion

A key aspect of physics is energy conversion. Mechanical energy—the sum of an object's potential and kinetic energy—can transform into different energy types when work is done. In the table jumping scenario, your kinetic energy at the moment of impact with the floor converts into heat and sound due to the air resistance and impact force. This phenomenon is the reason why your mechanical energy is not conserved. The same principle applies to the car crashing scenario; energy conversion occurs from the car's mechanical energy into thermal energy, sound energy, and energy used in deforming both the car and the tree.

Thermal Energy

Much of the lost mechanical energy in both scenarios transforms into thermal energy. Thermal energy is the internal energy present in a system due to the vibrational, rotational, and translational motions of its molecules. When a person lands on the floor, or a car slams into a tree, the noise and the damage observed are accompanied by the production of heat. This heat, dispersed into the environment, emanates from the energy dissipated by non-conservative forces like air resistance and the impact forces. It is this irretrievable dispersion of energy that leads to the conclusion that mechanical energy is not conserved when non-conservative forces are at play.