Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Lecture Notes

Transportation

Energy Use in Cars 2: ConstantSpeed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Question

• If we don’t burn gas, eventually the car would stop

• Friction and air resistance tend to oppose motion. We thus need energy to keep an object moving to over come these forces

Constant Speed Cruising

If a body in motion tends to stay in motion, why do we need to burn gas to travel at highway speeds?

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Background

1. Accelerating the car up to its cruising speed

2. Overcoming air resistance3. Overcoming rolling resistance4. Heat (partly converted to motion, flowing

to the environment with exhaust gases and by convection cooling of the engine)

Energy from the fuel in a car goes to 4 mainPlaces:

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Background

• As a car moves, it leaves behind a tube of swirling air (as in the figure below).

• The engine needs to provide the energy for all that swirling.

• We want to make a reasonably accurate estimate of how much energy we need for all the swirling air left behind.

Constant Speed Cruising

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BackgroundConstant

Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Model

• Imagine the swirling air is confined to a long tube region near the path of the car.

• Cross sectional area of tube = Atube

• Frontal area of car = Acar

• Acar < Atube

• Atube/Acar = Drag Coefficient, CD.• CD = 0.33 for a typical family sedan• CD = 0.9 for a cyclist

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Approach

• Determine how much energy the car loses to the air

• To do this, figure out the kinetic energy (K.E) of moving air.

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Approach

To figure out the K.E, we need:

• m (kg), the mass of the car• V (m3), the volume of the tube of air.• v (m/s), the velocity of car = velocity of air

Figuring out the K.E

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Approach

• d = distance traveled by the car = length of the tube of air that the car encounters

dLength

The total volume of the tube will be: dAtube gth)(Area)(LenVolume

And the mass of the tube will be:

dAtube Volume)(Density)(Mass

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Approach

So now the kinetic energy will be:

2mv21K.E

2

2

21

21

dvCA

dvA

Dcar

tube

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Approach

Given that the area of a typical family sedan is:

We calculate the work done against resistance, for each km a typical car driving at 50km/h travels

Calculating theWork Done Against Air Resistance

2m3)m5.1)(m2( A

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

ApproachCalculating theWork Done Against Air Resistance

2

21resistanceair against doneWork dvCDAcar

kJ 261/smkg 612,126

)m/s14)(m1000)(33.0)(m3)(kg/m3.1(21

22

223

So for each km travelled, 126kJ of work is done against air resistance

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

ApproachCalculating the Fuel Requirement, Per kmWe can calculate the fuel requirement using the efficiency formula:

InputEnergy Fuel OutputWork

Input Work OutputWork Efficiency

EfficiencyOutputWork InputEnergy Fuel

25%kJ 126

kJ 505

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

ApproachCalculating the Fuel Requirement, Per kmAnd to provide this amount of energy, we need to use:

litres of #Joules of #litreper Energy

litreper Energy Joules of #litres of #

MJL 32 kJ 505

km 1 drive toL 601.0

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Interpretation

• We need 0.016 L of fuel per km to overcome rolling resistance at 50km/h

• This is only 21% of the total energy cost of a car (0.076 L/km)

• Thus at this speed, air resistance is a very small part of the fuel requirement of the car.

• Since resistance changes with v2, it becomes a larger part of the fuel requirement at higher speeds ~0.064 L/km at 100 km/h

Constant Speed Cruising

Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia

Bibliography1. MacKay DJC. Sustainable Energy - Without the Hot Air (Online). UIT

Cambridge. p.262. http://www.inference.phy.cam.ac.uk/sustainable/book/tex/ps/253.326.pdf [25 August 2009].

2. Wikimedia Foundation Inc. Gasoline (Online). http://en.wikipedia.org/wiki/Gasoline [25 August 2009].

3. MacKay DJC. Sustainable Energy - Without the Hot Air (Online). UIT Cambridge. p.257. http://www.inference.phy.cam.ac.uk/sustainable/book/tex/ps/253.326.pdf [25 August 2009].

4. MacKay DJC. Sustainable Energy - Without the Hot Air (Online). UIT Cambridge. p.31. http://www.inference.phy.cam.ac.uk/sustainable/book/tex/ps/1.112.pdf [25 August 2009].

Constant Speed Cruising