0.25 Phy1220: relationships among kinematics, newton's laws, vectors  (Page 2/5)

In this case, the total mechanical energy is not conserved . Therefore, external forces are often referred to as non-conservative forces.

A quick review of internal forces

On the other hand, you also learned that if work is done on an object only by internal forces, the total mechanical energy possessed by the object cannot change. However, it can be transformed from potential energy to kineticenergy and vice versa.

In this case, the total mechanical energy is conserved . Therefore, internal forces are often referred to as conservative forces.

A quantitative relationship

The quantitative relationship between work and mechanical energy can be stated as follows:

MEf = MEi + We

where

• MEf and MEi represent the final and initial total mechanical energy possessed by the object respectively.
• We represents the work done on the object by external forces.

This equation states that the final amount of mechanical energy possessed by an object is equal to the initial mechanical energy plus the work done on theobject by external forces.

Potential energy plus kinetic energy

The total mechanical energy at any point in time can be the sum of potential energy (gravitational or elastic potential energy) and kinetic energy due to motion.

Given that, we can rewrite the earlier equation as:

KEf + PEf = KEi + PEi + We

where

• KEf and PEf represent the final kinetic and potential energy respectively.
• KEi and PEi represent the initial kinetic and potential energy respectively.
• We represents the work done on the object by external forces.

As mentioned, the work done by external forces can be either positive or negative work. Whether the work is positive or negative depends on the cosine ofthe angle between the direction of the force and the direction of the displacement of the object.

An ideal rocket example

Consider the following scenario. The owners of an experimental rocket lift the rocket onto a platform above ground level and set it up for firing.

Later, when they fire the rocket, it goes straight up while the rocket engine is burning. When the rocketengine runs out of fuel and stops burning, the rocket coasts to its apex and stops climbing.Then it falls back to the surface of the earth in an unglamorous free fall.

Simplifying assumptions

We will make some simplifying assumptions:

• The mass of the fuel is insignificant relative to the combined mass of the rocket and its payload. Therefore, expenditure of fuel doesn't affectthe mass of the rocket in a significant way.
• Air resistance is negligible. The rocket acts as if in a vacuum.

Initial conditions

Here are the initial conditions for the rocket experiment:

• Platform height = 15 meters.
• Mass of rocket and payload = 10kg.
• Thrust of rocket is constant at 150 newtons during burn.
• Burn time for the rocket = 10 seconds.

Legs of the trip

We will analyze the rocket's round trip from the ground, into the air, and back to the ground in several legs as described below:

• Leg A: Manually lifting the rocket from the ground to the platform.
• Leg B: Displacement of the rocket under rocket-engine power straight up.
• Leg C: Displacement of the rocket without power while coasting to the apex.
• Leg D: Displacement of the rocket in free fall from the apex back to the ground.