Projectiles, pendula, and special relativity

The kind of flat-earth gravity used in introductory physics appears in an accelerated reference system in special relativity. From this viewpoint, we work out the special relativistic description of a ballistic projectile and a simple pendulum, two examples of simple motion driven by earth-surface gravity. The analysis uses only the basic mathematical tools of special relativity typical of a first-year university course.Comment: 9 pages, 5 figures; to appear in American Journal of Physic

A CANADIAN POLICY ANALYST'S PERSPECTIVE

Agricultural and Food Policy,

Analytic approximations to the spacetime of a critical gravitational collapse

We present analytic expressions that approximate the behavior of the spacetime of a collapsing spherically symmetric scalar field in the critical regime first discovered by Choptuik. We find that the critical region of spacetime can usefully be divided into a ``quiescent'' region and an ``oscillatory'' region, and a moving ``transition edge'' that separates the two regions. We find that in each region the critical solution can be well approximated by a flat spacetime scalar field solution. A qualitative nonlinear matching of the solutions across the edge yields the right order of magnitude for the oscillations of the discretely self-similar critical solution found by Choptuik.Comment: 12 pages, Revtex, 9 figures included with eps

Ballistic trajectory: parabola, ellipse, or what?

Mechanics texts tell us that a particle in a bound orbit under gravitational central force moves on an ellipse, while introductory physics texts approximate the earth as flat, and tell us that the particle moves in a parabola. The uniform-gravity, flat-earth parabola is clearly meant to be an approximation to a small segment of the true central-force/ellipse orbit. To look more deeply into this connection we convert earth-centered polar coordinates to ``flat-earth coordinates'' by treating radial lines as vertical, and by treating lines of constant radial distance as horizontal. With the exact trajectory and dynamics in this system, we consider such questions as whether gravity is purely vertical in this picture, and whether the central force nature of gravity is important only when the height or range of a ballistic trajectory is comparable to the earth radius. Somewhat surprisingly, the answers to both questions is ``no,'' and therein lie some interesting lessons.Comment: 7 pages, 3 figure

Liberalisation and divestiture in the UK energy sector

Over the past 10 years, the government has privatised two energy industries — gas and electricity — and is presently selling British Coal. Vickers and Yarrow (1988) point out that the privatisation of utilities has two components, for the sale of assets to the private sector may be accompanied by changes in industrial structure. It is possible to sell assets without liberalisation, just as it would be possible to liberalise a market without asset transfers.2 The three energy industries were privatised with very different structures and competitive environments. This paper examines the structures chosen in the light of the benefits to government, private producers and consumers, focusing on whether restructuring and liberalisation should occur before or after privatisation. A similar choice exists after flotation between divestiture and restructuring of the industry itself and changing the external competitive or regulatory environment. Within energy, we show that the structure chosen for one industry affects the options available for another, because of the complex interactions within the sector.