**Problem : **
In the lab frame a particle has velocity (*v*_{x}, *v*_{y}) = (0.6*c*, 0) acceleration (*a*_{x}, *a*_{y}) = (2, 3). What force
is observed to be acting on the particle in the instantaneous inertial frame of the particle?

**Problem : **
If *p* = *γ*^{2}*mv*_{x}, what would *F*_{x} be?

= = = 2γ^{4}va |

Then:

F = = m(v + γ^{2}) = m(2γ^{4}v^{2}a + γ^{2}a) = maγ(2v^{2}γ^{2} -1) = γ^{3}ma(1 + v^{2}/c^{2}) |

**Problem : **
Use the definition of force found in Section
3 and what you know about the
relationship
between force and energy to find an expression for the relativistic energy.

Fdx = (γ^{3}ma)dx | |||

= γ^{3}mvdx | |||

= γmc^{2}|^{v2}_{v1} |

Which is *ΔE*
for a particle changing from *v*_{1}
to *v*_{2}
. Of course, the energy change is offset by
an equivalent change in potential energy such that total energy is conserved.

**Problem : **
Define a quantity *b*_{i} = . Let (*b*_{x}, *b*_{y}) = *γ*^{3}, *γ* similar to the relation for force. How does this quantity transform
between a stationary frame and a frame moving with speed *v* in the *x*-direction?

(b_{x}', b_{y}') = γ^{3}, γ^{3} = γ^{4}, γ^{3} |

Thus,

**Problem : **
In some frame *F'* which is moving at speed *v* in the *x*-direction with respect to frame *F*, there is a
mass attached the end of a spring. The spring is constraining to vibrate along a straight line which lies at an
angle *θ*' to the *x'*-direction in frame *F'*. In frame *F*, what is the direction of the spring
oscillation, the direction of the acceleration of the mass, and the direction of force on the mass?

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