Examples

Remember the intro video? Examples are there in order to give you some insight as to how the course leaders solve questions. It should help you get a grip on how to solve questions.

• The elementary ones should be able to be followed and solved by everyone.
• The medium ones can be a bit tricky in some cases, don’t worry if you don’t get them the first time.
• The advanced ones are there to challenge you and for more advanced learners.

Good luck!

EXAMPLE 2.1A: ENERGY FORMS (ELEMENTARY)

Energy comes in many forms: kinetic energy, potential from gravity, internal energy, bonds in molecules, etc. Here we are going to develop a feel for energies, by converting one into the other and by seeing what we get.
Consider a mass m of 10 kg. It is drop from a height of 100 m (without initial velocity). We will neglect friction from the air.

1. Compute the velocity with which the mass impacts on the earth.
2. If all this energy is finally converted into internal energy of the mass, i.e. heat, what will the temperature rise of the mass be? The specific heat is cp=2500 J/kgK.

Chemical energies, stored in bonds between molecules, is very large. Consider 1 gram of petrol. When combusted, it will release 45 kJ of energy.

1. What would the velocity of this 1 gram be if all combustion energy would be used as kinetic energy for the mass?
2. What height could it reach (assume gravity’s acceleration stays 9.81 m/s2 ‘for ever’)?

EXAMPLE 2.1B: ENERGY OF WATER FALLS (MEDIUM)

In the theory lecture we used the Victoria waterfall as an example to show various forms of energy. Here we will calculate how energy per second is associated with the falls.
The fall has a height of 108m. The average water flow rate is 1088 m3/s.

Assuming all potential energy can be converted to electricity, what would be the power of the “Victoria Power Station”? For reference: a standard modern fossile fule plant has a electric power output of 1000-2000 MWatt.

Similarly, caclulate the power of a hypothetical “Niagara Power Station”. The Niagara falls have a height of 51m and an average flow rate of 2400 m3

EXAMPLE 2.1C: MAKING A CUP OF TEA (ADVANCED)

Time for a cup of tea. So, you need to get boiling water. Easy enough, you take an water cooker with an (electrical) power of 2200 Watt. You poor in 0.5 liter of water from the tab. This water has a temperature of 15oC. We would like to know, how long it takes for the water to boil.

1. Set up an energy balance for the water in the cooker. Determine which kind of energy you would like to deal with. Is this a steady or unsteady balance?
2. Turn this energy balance into an equation for the temperature of the water. To make lif simple, we will approxiamte the problem by stating that the heat capacity of water is a constant with a value of 4200J/kgK. Moreover, we will compute when the temperature of the water has increased to 100oC, but does just not boil.
3. Solve the temperature equation and find how long it takes for the water to just/just not boil.

BONUS: EXAMPLE 2.1D: COMPARING ENERGY CONCENTRATIONS (ELEMENTARY)

In this exercise we are going to compare energy concentrations of various different cases. We will try to express the energy concentration in terms of energy per kg.

1. What is your kinetic energy per kg if you are running at 5m/s?
2. What is your potential energy per kg when on top of the Eiffel Tower in Paris (top is 324m high)
3. What is your kinetic energy per kg when flying in a Boeing cross Atlantic (cruise speed 900km/h); what is your potential energy per kg (altitude 10km)?
4. Figure out what the energy in regular CocaCola is per kg (try to look it up on a bottle or tin can)
5. Try to find the chemical energy stored in a kg of gasoline.
6. What is the nuclear energy released when a kg of hydrogen is fused completely to Helium (as happens in our sun)? Four hydrogen atoms are needed to form one atom of Heliem.

These numbers give you hopefully some feeling for energy densities: kinetic and potential are really small, chemical is hugh and nuclear ‘goes through the roof’.