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 continuously be converted back and forth between potential and kinetic energy without loss. This is an excellent visualization of energy that can be demonstrated with vpython. The spring will oscillate up and down constantly converting between Elastic Potential Energy and Kinetic Energy. 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 moving through space at constant velocity. It 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 is the proton's velocity?

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

Kinetic energy is very important is baseball. When a batter hits a ball, kinetic energy from the bat is transferred into the ball such that the ball flies out into the field. Assuming solid contact between the ball and the bat, the amount of kinetic energy transferred to the ball is directly proportional to the kinetic energy of the bat. As this is a non-relativistic application, there are only two variables to consider when calculating the kinetic energy of the bad: m and v. While a heaver bat would have greater kinetic energy than a bat of smaller mass traveling at the same velocity, heavier bats are more difficult to swing quickly and generally have lower velocity. Hence, batters must try to choose the a bat that they can swing very fast that still has reasonable mass to it to try to maximize kinetic energy ([math]\displaystyle{ \frac{1}{2}mv² }[/math]).

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