Introduction to Health Physics: Fourth Edition - Ruang Baca FMIPA UB

and at v = 0.99 c,
m = 9.11 × 10−31 kg
(0.99 c)2
1 −
c 2
= 64.6 × 10 −31 kg
REVIEW OF PHYSICAL PRINCIPLES 7
Example 2.1 shows that whereas an electron suffers a mass increase of only 0.5%
when it is moving at 10% of the speed of light, its mass increases about sevenfold
when the velocity is increased to 99% of the velocity of light.
Kinetic energy of a moving body can be thought of as the income from work put
into the body, or energy input, in order to bring the body up to its final velocity.
Expressed mathematically, we have
W = E k = f × r = 1
2 mv2 . (2.5)
However, the expression for kinetic energy in Eqs. (2.3) and (2.5) is a special
case since the mass is assumed to remain constant during the time that the body
is undergoing acceleration from its initial to its final velocity. If the final velocity
is sufficiently high to produce observable relativistic effects (this is usually taken as
v ≈ 0.1c = 3 × 10 7 m/s, then Eqs. (2.3) and (2.5) are no longer valid.
As the body gains velocity under the influence of an unbalanced force, its mass
continuously increases until it attains the value given by Eq. (2.4). This particular
value for the mass is thus applicable only to one point during the time that body
was undergoing acceleration. The magnitude of the unbalanced force, therefore,
must be continuously increased during the accelerating process to compensate for
the increasing inertia of the body due to its continuously increasing mass. Equations
(2.2) and (2.5) assume the force to be constant and therefore are not applicable
to cases where relativistic effects must be considered. One way of overcoming this
difficulty is to divide the total distance r into many smaller distances, r1, r2, ...,
rn, as shown in Figure 2-1, multiply each of these small distances by the average
force exerted while traversing the small distance, and then sum the products. This
process may be written as
W = f1 r1 + f2 r2 +····+fn rn
f
r 0
Δr1 Δr2 Δr3 Δr Δr
n-1 n
rn
r
(2.6a)
Figure 2-1. Diagram illustrating that the total work done in accelerating a body is W = n
fnrn.
n=1