Kinetic Energy

Kinetic Energy

Objects in motion have energy associated with them. This energy of motion is called kinetic energy. Kinetic energy, often abbreviated as KE, is usually given in the standard S.I. units of Joules (J). KE is also given in units of kilo Joules (kJ). [math]\displaystyle{ 1 kJ = 1000 J }[/math]. [math]\displaystyle{ 1 J = 1 kg*(m²/s²) }[/math]. Other types of energy include Rest Mass Energy and Potential Energy.

A Mathematical Model

The relativistic equation for kinetic energy according to Einstein's Theory of Relativity is [math]\displaystyle{ KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1) }[/math]. However, for cases where an object's velocity is far less than the speed of light ([math]\displaystyle{ 3X10^8 m/s }[/math]), one can use the simplified kinetic energy formula: [math]\displaystyle{ KE=\frac{1}{2}mv² }[/math]. In most cases the simplified kinetic energy formula gives a result with only minimal error. However, for near light speed calculations, such as those involving subatomic particles such as electrons, protons, or photons, the relativistic equation must be used. Usually we think of the simplified kinetic energy formula as the way to calculate the kinetic energy of an average object.

A Computational Model

By Conservation of Energy, energy can be converted but it cannot be created nor destroyed. Hence, in an isolated system, energy can be converted back and forth between potential and kinetic energy continuously without loss. This is an excellent visualization of energy that can be demonstrated with vpython. https://trinket.io/glowscript/87a35d5778

Examples

Simple

A ball is rolling along a frictionless surface at a constant [math]\displaystyle{ 19 m/s }[/math]. The ball has mass [math]\displaystyle{ 12 kg }[/math]. What is the kinetic energy of the ball in Joules?

Solution:

1. Because the ball's velocity is far less than the speed of light, we can use the simplified kinetic energy formula.
   
2. [math]\displaystyle{ KE=\frac{1}{2}mv² }[/math]
   
3. [math]\displaystyle{ KE=\frac{1}{2}(12 kg)*(19 m/s)² }[/math]
   
4. Hence, KE = [math]\displaystyle{ 2166 J }[/math] or [math]\displaystyle{ 2.166 kJ }[/math]
   

Middling

An electron is moving through space at a constant [math]\displaystyle{ 2.9X10^8 m/s }[/math]. The electron has mass [math]\displaystyle{ 9.1X10^-31 kg }[/math]. What is the kinetic energy of the ball in Joules?

Solution:

1. Because the electron's velocity is close to the speed of light, we must use the relativistic kinetic energy formula.
   
2. [math]\displaystyle{ KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1) }[/math]
   
3. [math]\displaystyle{ KE=(9.1X10^-31 kg)(3X10^8 m/s)²(\frac{1}{\sqrt{1-\frac{(2.9X10^8 m/s)²}{(3X10^8 m/s)²}}} -1) }[/math]
   
4. Hence, KE = [math]\displaystyle{ 2.38X10^-13 J }[/math] or [math]\displaystyle{ 2.38X10^-16 kJ }[/math]
   

Difficult

An proton is found to have kinetic energy [math]\displaystyle{ 2.38X10^-15 J }[/math]. The proton has mass [math]\displaystyle{ 1.67X10^-27 kg }[/math]. What velocity would the proton have to be moving at to have this kinetic energy?

Solution:

1. Because we are dealing with a subatomic particle, we should probably use the relativistic kinetic energy formula as the approximate kinetic energy formula may be very inaccurate if the particle is moving at a velocity near the speed of sound.
   
2. [math]\displaystyle{ KE=mc²(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1) }[/math]
   
3. [math]\displaystyle{ \frac{KE}{mc²}=(\frac{1}{\sqrt{1-\frac{v²}{c²}}} -1) }[/math]
   
4. [math]\displaystyle{ \frac{KE}{mc²} + 1=\frac{1}{\sqrt{1-\frac{v²}{c²}}} }[/math]
   
5. [math]\displaystyle{ \frac{1}{\frac{KE}{mc²} + 1}=\sqrt{1-\frac{v²}{c²}} }[/math]
   
6. [math]\displaystyle{ 1-\frac{v²}{c²}=(\frac{1}{\frac{KE}{mc²} + 1})² }[/math]
   
7. [math]\displaystyle{ \frac{v²}{c²}=1 - (\frac{1}{\frac{KE}{mc²} + 1})² }[/math]
   
8. [math]\displaystyle{ v²=c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²) }[/math]
   
9. [math]\displaystyle{ v=\sqrt{c²*(1 - (\frac{1}{\frac{KE}{mc²} + 1})²)} }[/math]
   
10. [math]\displaystyle{ v=c*\sqrt{1 - (\frac{1}{\frac{KE}{mc²} + 1})²} }[/math]
    
11. [math]\displaystyle{ v=3X10^8 m/s*\sqrt{1 - (\frac{1}{\frac{(2.38X10^-15 J)}{(1.67X10^-27 kg)(3X10^8 m/s)²} + 1})²} }[/math]
    
12. v = [math]\displaystyle{ 2.999X10^8 m/s }[/math]. This means that the proton is moving very close to the speed of light and hence our choice to use the relativistic kinetic energy equation was a good one.
    

Connectedness

1. How is this topic connected to something that you are interested in?
2. How is it connected to your major?
3. Is there an interesting industrial application?

History

Kinetic energy can be traced all the way back to Aristotle who first proposed the concept of actuality and potentiality (actuality being kinetic energy and potentiality being Potential Energy). The connection between energy and mv² was first developed by Gottfried Leibniz and Johann Bernoulli, whom described it as the "living force." William Gravesande tested this by dropping weights from different heights into a block of clay, discovering a proportionality between penetration depth and impact velocity squared. William Thomson is credited for devising the term "kinetic energy" in the mid 1800's.

See also

Potential Energy
Rest Mass Energy

Further reading

Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9

External links

http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy
https://en.wikipedia.org/wiki/Kinetic_energy

References

Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 9
http://www.physicsclassroom.com/class/energy/Lesson-1/Kinetic-Energy
https://en.wikipedia.org/wiki/Kinetic_energy