In mathematics, a rational number is any number that can be expressed as the quotient
or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q
may be equal to 1, every integer is a rational number. The set of all rational numbers,
often referred to as ”the rationals”, is usually denoted by a boldface Q (or blackboard
bold , Unicode ); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian
for ”quotient”. The decimal expansion of a rational number always either terminates
after a finite number of digits or begins to repeat the same finite sequence of digits over
and over. Moreover, any repeating or terminating decimal represents a rational number.
These statements hold true not just for base 10, but also for any other integer base (e.g.
binary, hexadecimal). A real number that is not rational is called irrational. Irrational
numbers include √2, , e, and . The decimal expansion of an irrational number continues
without repeating. Since the set of rational numbers is countable, and the set of real
numbers is uncountable, almost allreal numbers are irrational.
We study Logarithmically Spiral Trajectories and, in particular, we look for a solution to minimize the transit time of a Spacecraft propelled by a Solar Sail, while simultaneously minimizing the area of the Solar Sail, which would allow us to carry more payload on board. We start by analyzing the forces that act on the Spacecraft taking into account that its propellant is a Solar Sail; we use the studied forces to deduce the motion equations. We then solve this motion equation with a Runge-Kutta 4 method and transform the problem of minimizing time and area to a Non-linear Optimization problem. When solving the NLP we also try to minimize the relative final speed of th spacecraft with the destination planet in order to guarantee the possibility of a safe landing on its surface. The model improves when an angle parameter α (describing the angle formed by the Solar Sail with the colliding photons) is defined as a piecewise constant function and optimized whose values are optimized in every interval to minimize transit time and Area. Using the developed model to optimize the trajectory to be followed for sending from Earth to Mars a 2000kg-spacecraft propelled by a Solar Sail, subject to the condition that at trajectory start Mars and Earth were at their closest approach, and the Arrival Relative Velocity is less than 9km/s, give us a minimal transit time of 500days and a minimal area for the Solar Sail of 183158m2, meaning that the maximal payload would be 718kg. Compared with different number of partitions of α, the optimum stays stable. This gives a solid optimal trajectory and a great result for the numerical method used. Actually, waiting until the best moment to throw the Spacecraft, id est, Mars is at 1.14 radians respectively to Earth initial position, the minimal sail area 145950 m2 and, therefore, ables to transport until 978 kg of payload with the same transit time. In addition and to conclude we tried the model to optimize the inverse trajectory.
Marco Praderio Bova, Eneko Martin Martinez, & Maria dels Àngels Guinovart Llort
This report provides insight into the magnetic phenomenon of Hysteresis. Hysteresis is defined as a retardation effect where the magnetisation of a magnetic material lags behind the magnetizing force. Here we will explore the hysteresis loop for a silver steel ferromagnet and use this to discover it’s magnetic properties. The method used will be to place a ferromagnet inside a solenoid with an alternating voltage which will continually reverse the magnetic field and magnetism direction. The relation between these two quantities will be used to produce a hysteresis loop from which magnetic properties can be deduced. The results obtained were: saturation magnetisation = (8.4±0.5)(105)Am-1; remnant magnetisation = (5.9±0.5)(105)Am-1; coercive field: (4.3±0.5)(104)Am-1; energy expended per cycle per unit volume of material: (1.55±0.05)(103)Jm-3s-1; energy product: (8.7±3.0)(104)Jm-3.