(III) If a particle moves in the xy plane of system S (Fig. 36–12) with speed u in a direction that makes an angle θ with the x axis, show that it makes an angle θ' in S' given by $\tan θ^{'} = (\sin θ) \sqrt{1 - v^{2} / c^{2}} / (\cos θ - v / u)$ .

검증된 단계별 안내

Start by understanding the problem: The particle is moving in the xy-plane of a reference frame S' with speed u at an angle θ' to the x'-axis. We need to determine the angle θ that the particle's velocity makes with the x-axis in the reference frame S, which is moving relative to S' with velocity v along the x-axis. This involves using the relativistic velocity transformation equations.

Write down the relativistic velocity transformation equations for the components of velocity. For the x-component: u_x = (u'_x + v) / (1 + (u'_x * v) / c²). For the y-component: u_y = u'_y / γ(1 + (u'_x * v) / c²), where γ = 1 / √(1 - v² / c²).

Express the components of the velocity in S' in terms of u and θ': u'_x = u * cos(θ') and u'_y = u * sin(θ'). Substitute these into the transformation equations to find u_x and u_y in S.

The angle θ in S is given by tan(θ) = u_y / u_x. Substitute the expressions for u_x and u_y obtained from the previous step into this formula. Simplify the resulting expression to show that tan(θ) = (sin(θ') * √(1 - v² / c²)) / (cos(θ') - v / u).

Conclude by verifying that the derived expression matches the given formula. This involves ensuring that all terms are consistent with the relativistic velocity transformation equations and the geometry of the problem.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular reference frame. In this context, it is crucial to understand how the speed and direction of a particle change when viewed from different inertial frames, especially when considering the effects of motion at significant fractions of the speed of light.

Lorentz Transformation

The Lorentz transformation equations describe how measurements of time, length, and other physical quantities differ for observers in different inertial frames moving relative to each other at constant speeds. These transformations are essential for understanding how angles and velocities are perceived in different frames, particularly in the context of special relativity.

Trigonometric Relationships in Physics

Trigonometric relationships, such as sine and cosine, are fundamental in analyzing motion in two dimensions. In this problem, the angles θ and θ' relate to the components of velocity in the x and y directions, and understanding these relationships is key to deriving the expression for tan θ in terms of θ' and the velocities involved.