Page:TolmanNon1.djvu/3

Relation between the Components of Force Parallel and Perpendicular to the Acceleration.

Consider a body (fig. 1) moving with the velocity

${\mathsf {u}}=u_{x}{\mathsf {i}}+u_{y}{\mathsf {j}}.$

Let it be accelerated in the Y direction by the action of the component forces ${\mathsf {F}}_{y}$ and ${\mathsf {F}}_{x}$.

From equation (2) we have

${\mathsf {F}}_{x}={\frac {m_{0}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}{\frac {du_{x}}{dt}}+{\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\right)u_{x}$

(3)

${\mathsf {F}}_{y}={\frac {m_{0}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}{\frac {du_{y}}{dt}}+{\frac {d}{dt}}\left({\frac {m_{0}}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}\right)u_{y}$

(4)

Introducing the condition that there is no acceleration in the X direction, which makes $du_{x}/dt=0$, further noting that $u^{2}=u_{x}^{2}+u_{y}^{2}$, by the division of equation (3) by (4) we obtain

${\frac {{\mathsf {F}}_{x}}{{\mathsf {F}}_{y}}}={\frac {u_{x}u_{y}}{c^{2}-u_{x}^{2}}},$

${\mathsf {F}}_{x}={\frac {u_{x}u_{y}}{c^{2}-u_{x}^{2}}}{\mathsf {F}}_{y}$

(5)

Hence in order to accelerate a body in a given direction, we may apply any force ${\mathsf {F}}_{y}$ in the desired direction, but must at the same time apply at right angles another force ${\mathsf {F}}_{x}$ whose magnitude is given by equation (5).

From a qualitative consideration, it is also possible to see the necessity of a component of force, perpendicular to the