In Newtonian Mechanics, the Newtonian Kinetic Energy KE_(N) of a particle with mass m and velocity v is defined as
KE_(N)=(1)/(2)mv^(2)
In Einstein's Theory of Relativity, the Total Energy of such a particle is defined as
E=(mc^(2))/(sqrt(1-((v)/(c))^(2)))
where c~~3.0 imes 10^(8) meters/sec, and the Relativistic Kinetic Energy KE_(R) is defined as
KE_(R)=E-mc^(2)=mc^(2)((1)/(sqrt(1-((v)/(c))^(2)))-1)
a) Find the first four non-zero terms of the Maclaurin series for the function (1)/(sqrt(1-x)). Then, by substitution, find the first four non-zero terms of the Maclaurin series for the relativistic kinetic energy KE_(R). What do you notice?
b) Compare the Newtonian kinetic energy KE_(N) and the relativistic kinetic energy KE_(R) of a particle with mass m and velocity v by finding the ratio K(E_(R))/(K)E_(N) for (v)/(c)=0.1,(v)/(c)=0.5, and (v)/(c)=0.8. What do you notice?
defined as In Newtonian Mechanics,the Newtonian Kinetic Energy K EN of a particle with mass m and velocity u is
In Einstein's Theory of Relativity,the Total Energy of such a particle is defined as EE mc2 1-0/c2 where c ~ 3.0 108 meters/sec,and the Relativistic Kinetic Energy KER is defined as
KER=Emc2=mc2
a Find the first four non-zero terms of the Maclaurin series for the function Then,by substi- v1-x tution, find the first four non-zero terms of the Maclaurin series for the relativistic kinetic energy K ER What do you notice?
b) Compare the Newtonian kinetic energy K EN and the relativistic kinetic energy KER of a particle with mass m and velocity v by finding the ratio KER/KEN for v/c=0.1,v/c=0.5,and v/c=0.8.What do you notice?