Learn It: the Principle of Conservation of Energy

This has nothing to do with turning off your stereo when you leave a room. This fundamental piece of science says that you cannot destroy or create energy. (If you find a way – keep it quiet until you have taken out a patent – and you’ll be a rich person!)

People often say that you use up energy. That is misleading because it suggests that once it is used you can’t get it back. In fact, when you use energy, you are simply converting it into other forms of energy – so it is still out there: it is just no longer in the nice, useful form that you had it in a moment before.

Quite often if you can’t quite put your finger on where the energy has gone (i.e. what form you have turned it into) you have turned it into heat, which is lost to the surroundings.

Statement: The principle of conservation of energy states that energy can neither be created nor destroyed; it merely changes from one form to another.

or

In any closed system the total amount of energy is constant.

While the total amount of energy in any closed system does not change, the energy may be changed from one form to another. Not all forms of energy are equally useful to mankind.

When we talk of "energy shortages", we mean shortages of energy in readily usable forms - the total amount of energy in the Universe never changes. But while it is relatively easy to convert chemical energy (e.g. in coal or oil) to other forms (e.g. internal or electrical), it is very difficult to convert internal energy to other forms. Thus, when coal is burned much of its available chemical energy is converted to internal energy in the atmosphere and is effectively lost.

Demonstration: see an animation of Newton’s Cradle,demonstrating conservation of momentum and energy. (External link, use back button to return.)

Worked example:

Question 1
A stone of mass 500 g is thrown vertically upwards with a velocity of 15 m s-1. Find:  (a) the potential energy at greatest height;
(b) the kinetic energy on reaching the ground.

Solution:
Note: by the principle of conservation of energy the kinetic energy of the stone at the start will equal its potential energy at the top of its path. So, the potential energy at greatest height:

E_P =   E_k   =  ½mv²
=   ½ × 0.5 × 15²
Ep  =  56.25 J

In accordance with the principle of conservation of energy, the whole of this potential energy becomes transferred to kinetic energy when the stone reaches the ground again. Hence kinetic energy on reaching the ground = 56.25 J.

Answer.  P.E. at greatest height    = 56J 
              K.E. on reaching ground = 56J

 To solve the next problem it is useful to know the following:

Recall:

The momentum of a body is the product of its mass by its velocity p  =  mv

Law of Conservation of Momentum: the momentum before a collision = momentum after, provided no external forces act:
m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

Question 2. [Higher Level]

During a shunting operation, a truck of total mass 15 metric tonnes (t) moving at 1 m s^-1, collides with a stationary truck of mass 10 t.

If the two trucks are automatically connected so that they move off together, find their velocity.

Also calculate the kinetic energy of the trucks: (a) before and (b) after collision. Explain why these are not equal. (1 tonne = 1000 kg.)

Solution:

By the principle of conservation of momentum,
momentum before collision = momentum after collision

Let v = common velocity after collision, then using t and m s-1 units of momentum, 

(15 × 1) + (10 × 0)  =  (15 + 10) × v
  = >v  = 0.6 ms^-1

E_k    =    ½mv²    ( m in kg; v in m s^-1)
E_k   before collision      =   ½15 000 × 1² = 7500 J
E_k   after collision         =   ½ 25 000 × 0.6² = 4500 J

Answer: Velocity before collision = 0.6 m s^-1
E_k   before collision = 7500 J
E_k   after collision = 4500 J

In accordance with the principle of conservation of energy, the total energy after collision is the same as that before.

Before collision the whole of the energy is kinetic in the moving truck. But when collision occurs, part of this becomes transferred to internal energy in both trucks (k.e. and p.e. of molecules) and part into sound energy (k.e. and p.e. of air molecules). The remainder is left as mechanical kinetic energy in both trucks.

Consequently, mechanical kinetic energy after collision is less than mechanical kinetic energy before collision.

Note: You should now review: The practical method you used to measure the acceleration due to gravity, g.

Demonstration: see an animation of elastic and inelastic collisions. (External link, use back button to return.)

Next: Learn It - Energy conversions