Study Blog Part 4: Particle Kinematics

Even though I shouldn't have (being in school/class, trying to get work done lol) I watched PartinG´s Ro16 games. Wow. His games in the loser´s match and the final match were,.. Just wow!
Now, I am a Terran player. I don't really care for PvZ in the slightest, imo it is the most boring non-mirror match-up by far (when comparing the three; the race designs don't mix up well sadly) but watching PartinG perform his magical Immortal Push was amazing. I haven't seen it in a while, and I didn't think that I would see it again, but when PartinG delivers, he delivers!

The final match was worth watching as well. The sheer mind games were crazy! His collossi push was incredibly smart and his openings in G2 and G3 were really clever. Too bad his follow-up plan on Whirlwind was a bit coin flippy (I'm guessing he assumed that Life would take a quick 3rd and play it out normally?) but seeing him turn it around on Akilow Flats was pretty sweet. Imo, watching mass mutas in PvZ is one of the most frustrating things in SC2 that I can think of, and to see PartinG pull through that win was satisfying. Sure, Life maybe didn´t play as well as he should there (mutas engaging while being stormed, no flank, no tech behind heavy harrass) but PartinG played well!

i) Topics covered

Coordinates of motion

ii) Time spent/what sessions?
Reading theory = 2h
This week´s exercises =4h
Catching up = 2h
Exercises to be handed in = 2h
Repetition = around 2 h

iii) Topics explained

This week is the first week that we deal with particles in motion. Earlier, with statics, we have done calculations on rigid bodies which have been in rest.

Coordinates of motion deals with the question of where an object will be given that we know where they are located, how they are moving and how they are being accelerated. It doesn't deal with the question of how the objects originally were acted upon by exterior forces.

Motion of objects can be a complex matter, hence is why we use three different ways to describe them: we can describe motion of a particle with cartesian coordinates, with natural components and with cylindrical coordinates.

Cartestian coordinates uses the standard x-y-z-axis system to describe a particle's position as a function of time. This is useful, because we have find the derivative of the position with respect to time, hence the velocity. The same can be done to the velocity to find the acceleration of the particle. The standard right-axis coordinate system provides a simple way to describe position and motion of objects.

Natural components is another useful tool when describing motion. It relies on two components; one that points in the same direction as the velocity and one that points in the direction of the normal to the velocity. For non-constant velocities, this of course means that the directions of the components are not constant, but can be explained by a function for the position which relies on the length traveled from an arbitrary position, which in turn depends on time. This would mean that r : r(s(t)), where r is the position of the particle, s is the length of the path taken and t is time. The velocity and acceleration is found by differentiating with respect to time. See below!

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Cylindrical coordinates deal with motion which move in a circular shape. Polar coordinates is usually standard here. The two (in 2D) coordinate vectors used points in the direction of the particle's position from the origin and the other one points in the direction perpendicular to the first vector (determined by right hand rule). With some knowledge in calculus, we can from this find the velocity and acceleration of the particles, described in cylindrical coordinates.

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Generally, choosing an appropriate coordinate system is very important when solving physical problems, and mechanics is no exception. Whether the issue at hand is to simplify a system of forces acting on a body or having a complex system with several angles, the pathway to the solution of the problem, can become so much easier to see if a proper coordinate system is chosen straight away. In particle kinematics, we have a few more variants of coordinate systems than the standard cartesian system, and the reason being because it the cartesian system doesn't adequately explain every situation well enough to be the only go-to system. With the help of the right coordinate system, finding the solution becomes easier!

iv) What challenges did you face?

Calculus is used a lot in this section, which can be intimidating at first. The reason being because the derivatives to the unit vectors are not obvious at first, so the velocity and/or acceleration may look very different from the formula for the displacement of the particle.

I had a rather specific problem as well. I stumbled upon this problem before as well and still haven't found a solution to it. When acceleration is described as a function of velocity (e.g. a = -k*v*v), integrating it to find a distance, for example the distance traveling to go from an initial velocity to zero. The problem gets even more complex if acceleration has a constant as well! I'll have to look some more into it...

i) List of exercises
Finally, some exercises this week! I won't list the questions here before solving them, and the reason is twofold. Firstly, most importantly, I don't really wanna break any copyright rules, or risk breaking any rules. Also, the solution is structured in such a way that the question is stated in it (at least, that is my intention!!)
ii) Solving problems

Please, excuse my poor hand writing

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Define the velocity and acceleration of a particle.

Derive the expression of acceleration for a particle in cylindrical coordinates.

Explain the physical meaning of the tangential acceleration and the normal to the direction of the
acceleration when using the normal component system.

i) What topics? What do you know about this?
Next week is only scheduled for cylindrical coordinates, so I assume that it is also scheduled for some revision, which is good, because I need the extra time to catch up after last week!

I´ll probably just end up spending the extra time to catch up in calculations and to finish up the exercises which is to be handed in. The first test is coming up soon as well, but it should be doable, if I manage to do most of the exercises I missed.

Strange, looking for his account just to find out informed me that he has been temp banned Be brave our Italian stallion!