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Chapter 6

Q. 6.8

A relativistic particle P, initially at rest at the origin in frame  \psi,  is moving along a straight line under a constant force  \mathbf{F}_0.  Determine the relativistic speed and the distance traveled by P as functions of time.

Step-by-Step

Verified Solution

The equation of motion for the relativistic particle is given by (5.34) in which  \mathbf{F}(P, t)=\mathbf{F}_0  is a constant force and (6.9) is to be used. Hence, separation of the variables and integration of  \mathbf{F}_0 d t=d(m \mathbf{v})=d\left(\gamma m_0 \mathbf{v}\right),  with the initial values  \mathbf{v}(P, 0)=\mathbf{0}  and  \gamma=1,  yields  m \mathbf{v}=\mathbf{F}_0 t.  Thus, recalling (6.9) and noting that  \mathbf{v}=v \mathbf{t} \text { and } \mathbf{F}_0=F_0 \mathbf{t}  are parallel vectors, we have only one nontrivial component equation:  m_0 v /\left(1  –  v^2 / c^2\right)^{1 / 2}=F_0 t.  This scalar equation yields the rectilinear, relativistic speed

\mathbf{F}(P, t)=\frac{d \mathbf{p}(P, t)}{d t}=\frac{d}{d t}[m(P) \mathbf{v}(P, t)]                    (5.34)

m=\gamma m_0=\frac{m_0}{\sqrt{1  –  \beta^2}} \quad \text { with } \quad \beta \equiv \frac{\dot{s}}{c} \text {. }             (6.9)

v(P, t)=\frac{c k t}{\sqrt{1  +  (k t)^2}} \quad \text { with } \quad k \equiv \frac{F_0}{m_0 c} \text {. }                 (6.25a)

Introducing  v=\dot{s}  into (6.25a) , separating the variables, and integrating  d s=v d t  with the initial value  s(0)=0,  we obtain the rectilinear distance traveled by P:

s(P, t)=\frac{c}{k}\left(\sqrt{1  +  (k t)^2}  –  1\right) .             (6.25b)

Notice in (6.25a) that  v / c<1  for all  t, and  v / c \rightarrow 1 \text { as } t \rightarrow \infty;  that is, under a constant force, the relativistic particle speed cannot exceed the speed of light c. This  result is quite different from the corresponding speed  v=F_0 t / m_0  described by (6.22) for a Newtonian particle of mass  m=m_0  initially at rest and subject to a constant force  F_0;  in this case  v \rightarrow \infty  with t. If m_0 c  is large compared with  F_0 t  so that  k t \ll 1,  then (6.25a) and (6.25b) reduce approximately to

\mathbf{v}(P, t)=\frac{\mathbf{F}_0}{m} t  +  \mathbf{v}_0           (6.22)

v(P, t)=c k t=\frac{F_0}{m_0} t, \quad s(P, t)=\frac{1}{2} c k t^2=\frac{F_0}{2 m_0} t^2            (6.25c)

The se are the Newtonian formulas described by (6.22) and (6.23) for the  corresponding rectilinear motion of a particle of mass m_0  initially at rest at the origin and acted upon by a constant force  F_0.  In the present relativistic approximation , however, these results are valid for only a sufficiently small  time for which  v / c=k t \ll 1.

\mathbf{x}(P, t)=\frac{\mathbf{F}_0}{2 m} t^2  +  \mathbf{v}_0 t  +  \mathbf{x}_0                    (6.23)