CK-12-Physics - Intermediate

http://www.ck12.org Chapter 22. The Special Theory of Relativity

Illustrative Example 23.2.2

Show that for the casev<<cthe equationE=mc^2 +KEreduces toKE=^12 mv^2.

Solution:

Using the binomial expansion for√^1
1 +vc^22

whenvc,Ecan be expressed as follows:

E= mc
√^2
1 −vc^22

≈mc^2

(

1 +v

2

2 c^2 +...

)

, where additional terms in the expansion add smaller and more negligible terms,

and so are ignored.

Substituting the above approximation forEin the equationE=mc^2 +KE, we have

mc^2

(

1 +

v^2

2 c^2

+...

)

=mc^2 +KE→mc^2 +

1

2

mv^2 =mc^2 +KE→

KE=

1

2

mv^2

Check Your Understanding

True or False: The equation for relativistic kinetic energy for an object of massmtraveling with speedvcan be found
by substituting√m
1 −vc^22

for the massminto the equationKE=^12 mv^2.

Answer: False. We can see from the equationE=mc^2 +KEand Example 23.2.2 that the relativistic form for
kinetic energy is not reached in the same way as the relativistic equations for force and momentum.

Check Your Understanding

a. How much energy is released if a proton of mass 1. 67 × 10 −^27 kgis completely converted to energy?

Solution:

Using the rest energy of a proton, we have

E=mc^2 = ( 1. 67 × 10 −^27 kg)( 3. 00 × 108 )^2 = 1. 50 × 10 −^10 J

b. Express your answer in electron-volts(eV).

Answer: Recall that 1. 00 eV= 1. 60 × 10 −^19 J, therefore

1. 50 × 10 −^10 J


2. 60 × 10 −^19 J=^0.^9375 ×^10

(^9) eV or 938 MeV
Relative velocity
We began our discussion of relativity by stating that the results of Maxwell’s equations, that light must always travel
at a constant velocitycin a vacuum, contradicted the results of Galilean relative velocity. Namely, relative velocity
could not be calculated by the vector addition of velocities, that is~vr=~v 1 +~v 2. We have seen that in order to preserve
Galilean relativity, the expressions for both time and space needed to be changed. Since observers in different inertial
frames measure different times and different lengths, the equation for the calculating Galilean relative velocity must
also change. Einstein showed how the Galilean equation for relative velocities must be changed in order for light
(and the theory of electromagnetism) to remain consistent with Galilean relativity.
The correct relativistic addition of velocities equation which Einstein (and Lorentz) derived looks like this:vr=
v 1 +v 2
1 +v^1 c 2 v^2 ,