Chapter 1: Translational motion (C6913765)

 #toc { border: 1px solid #bba; background-color: #f7f8ff; padding: 1em; font-size: 90%; text-align: center; } #toc-header { display: inline; padding: 0; font-size: 100%; font-weight: bold; } #toc ul { list-style-type: none; margin-left: 0; padding-left: 0; text-align: left; } .toc2 { margin-left: 1em; } .toc3 { margin-left: 2em; } .toc4 { margin-left: 3em; } Last modified: 2138d agoWord count: 5,752 wordsLegend: Key principles // Storyline

1 Dimensions

Dimensions are the minimum measurements required to definea point. Minimization means dimensions should be independent of another, thereby maximizing efficiency of measures. Essentially, they are planes at right angles with another. Einstein proposed the fourth dimension, which is time. The SI unit of time is “seconds”.

“When model agencies collect height and waist, these dimensions are independent, as height cannot be defined with a waist measurement, or vice versa,” Jamie started, “This is because height is measured vertically, and waist is measured horizontally. Time is similarly independent.”

“For example, we can say Selena Gomez is 1.65m, or that the Wizards Movie runs for 98 minutes,” Mandy continued.

“You know how in 1 Cor 2:14, it talks of those without the Spirit interpreting things of God as foolish? It’s like the Spirit is its own dimension,” Mandy continued, “imagine a person from a 3D world sticking his fingers into a whiteboard where 2D people live. What the 2D people see would just not make sense.”

“The Holy Spirit does so much,” Mandy commented, “He formed the universe (Genesis 1:2-3), provides revelation (1 Corinthians 2:9-13), inspired scripture (2 Peter 1:20-21), regenerates (John 3:5-7), baptizes (1 Corinthians 12:13), indwells (Romans 8:9), assures salvation (Romans 8:14-16), enables Spiritual living (Galatians 5:16-25), bestows spiritual gifts to build up the church (1 Corinthians 12:4-11), and fruits (Galatians 5:22-23).

 Frequently asked questions When you talk about minimization, what are you minimizing?Measurements.Oh, so that's why they're at $90^{\circ}$ to another?Yes, because neither can be used to define the other. Think about it, no changes in the x-axis mean anything in terms of the y-axis, and vice versa.What's a plane?A plane is a flat 2D surface. Think of it like a piece of paper.

 Formative learning activity Maps to RK1.1 Identify 3 examples of dimensions, and explain why each are a dimension.

2 Vectors, components

Scalars are measurements with magnitude/power [but no direction]. Vectors have both magnitude and direction. Magnitude refers to the size property of an object. An example of a scalar quantity is time [which doesn’t have direction], and of vector is force [which has direction].

“So a scalar could be a mass 50kg or energy 100J,” Mandy said, “I’m not saying ‘50kg down’ or ‘100J east’.”

“In contrast, a vector could be force 500N down, or momentum 550sN up,” Mandy continued.

“So One Direction is a vector?” Blaire giggled.

“So long as the one direction you’re looking in is towards Christ ,” Mandy winked.

 Frequently asked questions What is "magnitude"?Strength, power, size: these are all appropriate synonyms used to describe magnitude.Remember Diglet from Pokemon? One of his moves was "magnitude". On the game console, the power of this move would be randomly generated, as a number between 1-10; the higher the number, the more powerful the attack, and the more damage it does on the opponent. Same idea here.Both scalars and vectors have magnitude. This is self evident, because without magnitude... there is... nothing [to talk about]!I see, so this is clearly different from "direction", which is not "power" (as in magnitude), but as in...Direction!Only vectors have direction. Scalars don't.What exactly is force and momentum?We'll get to that soon You may have heard about these principles in simple English, but they are principles which prerequisite the knowledge about... vectors!

 Learning activity Define the difference between a scalar and vector.Identify 3 examples of scalars, and 3 examples of vectors. Provide explanation for why each of the 6 are a scalar or vector respectively.

Vectors can be shown graphically as an arrow depicting the direction, connecting an initial point ($A$) with a terminating point ($B$), denoted by ($\vec{AB}$), with an arrow going from $A$ to $B$ from the top. The length of the [graphical] arrow depicts its magnitude. The tip of where the arrow is pointing to is called the head, and the opposing [starting] point is the tail.

[img]example-vector.png[/img]

 Frequently asked questions The starting side of the arrow is called the...TailThe ending side of the arrow, where the arrow points to, is called the...Head

 Learning activity Robin Hood shoots an arrow at $45^{\circ}$ elevation above the ground, towards the RHS of the page. Draw this.

Vector resolution

A vector can be resolved into its components, which are lines at $90^{\circ}$ to another, of which the sum is equal to the original vector. For instance, the red vector can be resolved into its x-component and y-component:

[img]vector-resolution.png[/img]

 Frequently asked questions What are components?Parts, constituents, making-up-the-larger-thing: these are all appropriate synonyms of the word "component".What is resolution?Where the larger-thing, is expressed instead, in its parts/constituents/etc.So in court, a resolution is where the parties reach a pre-court...Different type of resolution we're talking about here Same spelling, different word.A better analogy for resolution, is a computer screen's resolution. The higher the resolution, the greater the number of parts that make it up, and therefore the more smooth it looks. A very high resolution display is the iPhone's retina display, where you cannot even make out the individual pixels.So in the analogy, the pixel is the equivalent of the...ComponentSo another example is the components you find inside a computer. It's what makes it up [thereby determining its processing speed, read/write speed, graphics ability, and so forth].

Usually, this is defined as a graph with an x- and y-axis, expressed in the format $[a,b]$ where $a$ is the x-axis measurement, and $b$ is the y-axis measurement. Because vectors have direction, they need to be described with two numbers, whereas scalar quantities can be described with one.

“For example, a vector could be 100m north or 20m south, that is, it has direction!” Mandy said, “And it can be represented as $[0,100]$ and $[0,-20]$ respectively.”

The components of a vector can be determined using the Pythagorean theorem, which states that in a right angled triangle, $O^2+A^2=H^2$, where $O$ and $A$ are the two sides of the triangle at right angles with another, and H is the hypotenuse, which is the longest side in a triangle. When adding components, also useful will be trigonometry, namely $sin(\theta)=\dfrac{O}{H}$, $cos(\theta)=\dfrac{A}{H}$, and $tan(\theta)=\dfrac{O}{A}$, where $\theta$ is any angle in the triangle, $H$ is hypotenuse (just defined), $O$ is the side [of the triangle] opposite to the angle, and $A$ is the side adjacent to the angle. These trigonometric rules can be memorized using the mnemonic “Soh Cah Toa”.

[img]pythagorean-triangle.png[/img]

Jamie met his unlikely clinger girlfriend, Sophie Graham MD, the resident blonde klutz of the princess clique, at the Baptist church.

“Hey!” Sophie asked, “Do you know where QE Hospital is?”

“Not sure,” Jamie said grabbing his z-Phone to check.

Earlier, Jamie was sitting several pews behind Sophie, and had thrown a pencil in Sophie’s direction to try to grab her attention, and well, thought that perhaps this was her way of reciprocating her feelings. Little did Jamie know that Sophie knew nothing of the pencil-throwing escapade, but genuinely and independently, had feelings for him.

“Say if that pencil was thrown 60kmph at an 30° elevation, what are its components?” Mandy asked.

“By SOH-CAH-TOA,” Jamie responded, “The vertical component is $60.sin(30^{\circ})=30kmph$, and the horizontal component is $60.cos(30^{\circ})=52kmph$.”

“Oh! You can also check this is correct with the Pythagorean Theorem, that $\sqrt((30)^2+(52)^2)=60kmph$: QED,” Mandy remarked, “QED, Latin for quod erat demonstrandum, meaning I’m awesome aka demonstrated what I wanted.”

 Frequently asked questions How does the x- and y-axis relate to the concept of components?Any point in 2D, can be referred to as an x- and y-axis, in the form of [x,y], in relation to its origin (0,0).So how do you determine this x- and y- number? Usually, you're provided with a magnitude, and an angle?You can use a combination of either the Pythagorean theorem and/or trigonometry.What's the Pythagorean theorem?It states that in a right angled triangle, the square of its two sides, is equal to the square of the hypotenuse.What is the hypothenuse?The longest side of a right angled triangle. It's also the side opposite the right angle.Okay, so what's trigonometry?It is a formula that can be used, that doesn't require a right angled triangle. So it is more versatile than the Pythagorean theorem.The words "opposite" and "adjacent", what agnles are these exactly "opposite" and "adjacent" to?In pythagorus, they refer to sides, and the way they are assigned make no difference.In trigonometry, theta is an unknown angle, which can be arbitrarily assigned. So this is the theta that is present in the equation.

 Formative learning activity Maps to RK1.2 What are vectors? What about components?

The distinction between scalar and vector quantities is important, as vector addition cannot be achieved by simply adding the magnitude numbers together of the vector. Note also, that you cannot add vectors with scalars.

“What is 1m plus 1m?” Jamie asked.

“Well that’s 2m: easy pea-see!” Mandy replied.

“Hold your horse! How about if you strutted 10m down a catwalk, and back, how far have you travelled in total?”

“Sure,” Jamie laughed, “dancing 10m forth, 10m back.”

“20m, because 10+10=20,” Mandy replied.

“That’s right Mandy, that’s distance. But how far have you effectively travelled though, from start to finish?”

“0m, because I’m back where I started!” Mandy said.

“Right, and that’s displacement: your net movement.”

“You’re back to where you started, so that’s… 0m.”

“My goodness ,” Jamie laughed, “The world isn’t round, the moon is made of cheese, and 1+1=0 !!”

Vector addition can be achieved either graphically, or by addition of components. The graphical method involves attaching the vectors tail-to-head with tail-to-head, with the resultant vector going from the very tail to the very head. For instance, if we add 4N east with 3N south, graphically:

It makes sense that the resultant vector (in red) has a magnitude of 5N, because created, is a right angled triangle, and as a common right angled triangle, we know sides are 3, 4, and 5, which is the solution.

This can be quite complicated, especially when dealing with 3D vectors. The component method involves resolving a vector into its components, and then adding its components.

“Then, writing a vector in the form of [x-component, y-component, z-component], to add [1,2,3] and [5,5,5], this is [1+5,2+5,3+5]=[6,7,8], not requiring complex geometry,” Mandy commented.

Vector addition is possible graphically, because $\vec{A}+\vec{B}=\vec{A+B}$.

The order of the vector addition is irrelevant, meaning $\vec{A}+\vec{B}=\vec{B}+\vec{A}$.

Vector subtraction involves adding the converse/negative of the subtracted vector, or alternatively, $\vec{A}-\vec{B}=(\vec{A})+(\vec{-B})$. For example, if we subtract 4N east with 3N south, graphically:

[img]vector-subtraction-example.png[/img]

 Frequently asked questions What is vector subtraction?Like standard subtraction, it's the opposite of addition.So how's that exactly done?You add the subtracted vector. So you flip the subtracted vector around the opposite direction, and add that.

The most straightforward type of vector multiplication is scalar multiplication, where a vector is multiplied by a scalar. This doesn’t alter the direction, but rather, only the magnitude. As such, the magnitude is multiplied directly in using the component method, or the vector is repeated by the magnitude number of times graphically. For example, if the pink vector is multiplied by 3, the resulting blue vector is in the same direction:

[img]scalar-multiplication-example.png[/img]

Technically, a vector cannot be multiplied with another vector. However, two artificial means of vector-vector multiplication have been created–

Dot product creates a scalar, and is $a \cdot b=ab.cos(\theta)$, where $\theta$ is the angle between vectors $a$ and $b$. Dot product provides the multiplication of one vector, on to the projection of another onto the former vector. Thus, as perpendicular vectors have no projection on each other, the dot product is zero. For example, work $W=F \cdot s=|\vec{F}||\vec{s}|.cos(\theta)$. In other words, if $F=50N$ and $s=2m$, and both are in the same direction (i.e. $\theta=0$), the answer is simply $100J$. So, only the portion of the vectors that are parallel to another are used in the calculation of multiplication.

[img]dot-product-example.png[/img]

Cross/vector product creates a vector, and is $a \times b=ab.sin(\theta)$, where $\theta$ is the angle between vectors $a$ and $b$. Cross product is the opposite of dot product, multiplying one vector, on to the portion of another vector that is perpendicular to the former. The resulting vector is one that is perpendicular to both vectors $\vec{a}$ and $\vec{b}$, whose direction can be determined by the right-hand rule, such that if the index and middle finger [of the right hand] is placed in the direction of the crossing vectors, the resultant vector direction is in the direction of the thumb. If the two vectors are parallel to each other, the cross product is zero. For example, moment $M=\vec{F} \times \vec{s}=|\vec{F}||\vec{s}|.sin(\theta)$. In other words, if $F=50N$ and $s=2m$, and both are at $90^{\circ}$ to another, the answer is simply $100Nm$. So, only the portion of the vectors that are perpendicular to another are used in the calculation of multiplication. Say if the moment/spin produced is clockwise, this also needs to be included. Therefore, the solution would be 100Nm clockwise.

[img]cross-product-example.png[/img]

 Frequently asked questions Can you multiply a vector with a scalar?Yes, that's called scalar multiplication. It's where a vector is multiplied in that original direction, by the "scalar" number of times.Can you multiply a vector with another vector?Not really. But we have created 2 artificial means, including dot product, and cross product.What is dot product?It is a calculation that produces a scalar, namely, $ab.cos(\theta)$. Essentiallly, it seeks the projection of one vector onto another. In other words, it considers only the portion of the vectors which are parallel to another.What is cross product? Vector product?Cross product and vector product is the same thing. It is a calculation that produces a vector, namely, $ab.sin(\theta)$. Essentially, it is the opposite of dot product, considering only the portion of the vectors which are perpendicular to another.So you said the cross product produces a vector. Has it got a direction?Yeah it does! This is determined by the right-hand rule, such that the index and middle fingers [of the right hand] is placed in the direction of the crossing vectors, and the resultant vector is in the directon of the thumb.So as you can see, the cross product is perfectly perpendicular to all crossing vectors.How can vector multiplication identify parallel and perpendicular vectors?When cross product is zero, the vectors are parallel. That's because cross product considers only the perpendicular portion, so when 0, there must be no perpendicular portion.When dot product is zero, the vectors are perpendicular. That's because dot product considers only the parallel portion, so when 0, there must be no parallel portion.

 Formative learning activity Maps to RK1.3 What is vector addition?

4 Speed, velocity

Displacement is the vector version of distance [which is scalar]. Speed is defined as $\dfrac{distance}{time}$, and is hence scalar. Velocity is defined as $\dfrac{displacement}{time}$, and is hence the vector version of speed. Displacement is the shortest/effective distance from the initial to final position, whereas distance is the actual path taken. The SI unit of distance is “meters”.

“If you fly from London to NYC, the plane doesn’t fly in a straight line! It twists and turns,” Jamie said.

“So even though the displacement from London to NYC is 5585km,” Mandy said, “I’m going to be travelling a greater than this, because of the twists. The total distance I’ve travelled is referred to as distance.”

“The effect is that though speed may be a non-zero number, because velocity uses displacement, if we travel from London to NY and back, velocity will be 0m/s.”

[img]difference-between-distance-and-displacement.png[/img]

Note that velocity is a property that doesn’t depend on quantity, known as an intensive property. In contrast, a property that does depend on quantity is known as an extensive property, for example, mass and energy.

 Frequently asked questions What's the difference between displacement and distance?Displacement is a vector, distance is a scalar. Displacement is the shortest distance from 2 points, whereas distance is the actual path taken.What's the difference between velocity and speed?Velocity is a vector, speed is a scalar. Velocity uses displacement, and speed uses distance. Both have the same structure $s=\dfrac{d}{t}$, however.What's the difference between an intensive and extensive property?Intensive properties don't depend on quantity, whereas extensive properties do [depend on quantity]. Speed for example, doesn't increase simply because there is more of that object. Mass however, does.

 Formative learning activity Maps to RK1.4 What is speed? What is velocity, and how does is it distinct from speed?

5 Acceleration

Acceleration is $\dfrac{\Delta velocity}{time}$, where the Greek letter delta $\Delta$ means “change in”. For example, $60 kmph^2$ means that you are increasing your speed by $60kmph$ every hour, meaning you would take an hour to reach . In English, acceleration refers to positive acceleration, and deceleration refers to negative acceleration.

“Well yeah, of course, that’s the speed cars travel on the highway,” Mandy responded.

“Okay, how about if we went at $100 kmph^2$?”

“Hmm… acceleration is a bit harder,” Mandy thought.

“That’s why it’s easier to think of acceleration as change in velocity, per unit time.”

“So $100 kmph^2$ is gaining from 0kmph to 100kmph – every hour, which is INCREDIBLY slow!!!”

 Frequently asked questions What is acceleration?The change in velocity.What is deceleration?It refers to acceleration that is negative. That is, a reduction in velocity.

 Formative learning activity Maps to RK1.5 What is acceleration?

6 Free falling bodies

Free falling is the permission of an object to be subject to gravitational acceleration. Gravity is an acceleration downwards of $10m/s^2$ (in physics, the figure more generally used is $9.8m/s^2$).

“Newton founded gravity, because one of ‘em apples dropped on his head, and he wondered: why?” Mandy said.

“How ‘bout them apples,” Blaire giggled.

“I think I now know why I like red apples,” Mandy said, “them apples got product placement in Snow White .”

 Frequently asked questions What is free falling?Permitting an object to be subjected only to gravity.Gravity is dropping down?It is specifically acceleration downwards of $10m/s^2$.

The formula for determining instantaneous velocity is $v=a.t$. Based on the acceleration of $10m/s^2$, each second, an object is $10m/s$ faster than the preceding second. Thus, after 1 second, an object travels at $10m/s$; after 2 seconds, $20m/s$; after 5 seconds, $50m/s$; etc.

Where the object is not dropped, for example, thrown up at $50m/s$, gravity will still act, but in the opposite direction (at least at the start). The time taken for the object to reach its maximum height is 5 seconds, as the object slows $10m/s$ each second. The time taken for it to reach its maximum, back to its original height, is that time multiplied by two. So for an object thrown up at $50m/s$, it will take 5 seconds up, and 5 seconds down, so 10 seconds in total.

Distance travelled is the average velocity multiplied by time, and when acceleration is constant, average velocity is $v_{av}=\dfrac{1}{2}(v_{0}+v_{t})$. So for an object dropped, after 5 seconds, it would reach $50m/s$. The initial velocity was 0m/s (as it was dropped), meaning the average velocity would be $25m/s$. The time taken is $5 \times 25=125m$.

Time taken is more difficult to determine, and may require the uniform acceleration motion equation, which are equations that apply when acceleration is uniform [or constant], linking displacement ($s$), initial velocity ($v_{0}$), final velocity ($v_{t}$), acceleration ($a$) and time ($t$), and include:

• $s=s_{0}+v_{0}t+\dfrac{1}{2}at^2$
• $v_{t}^{2}=v_{0}^{2}+2as$, or in-substituting, $v^2=2gh$, if there is no initial velocity. As all potential energy is converted into kinetic energy, this formula can also be derived from equating PE and KE, namely $mgh=\dfrac{1}{2}mv^2$, and reducing to get the supra formula
• $v_{t}=v_{0}+at$

“I can’t seem to find it,” Jamie replied; realizing his stoush at Sophie was, well, free falling.

“By the way,” Jamie said, “my name is Jamie.”

“My real name is Sophia, but people call me Sophie.”

“What’s with the massive Disney smile ?” Jamie asked.

“Do I look like the girl from Tangled?” Sophie giggled.

Realizing that he couldn’t just continue fidgeting his phone, pretending to find a non-existent hospital, he gathered himself, preparing his next statement.

“Hey, did you get that pencil?” Jamie asked.

“What pencil?” Sophie remarked, confused.

And that was it. Jamie realized it was just all in his head; she was not interested in him, and just genuinely wanted to find this hospital.

“I can’t find it,” Jamie sighed, trying to figure if there was something he could ask her, just so this magical moment could be prolonged. But he had nothing. Or at least nothing came out.

“Alright,” Sophie replied after a moment of silence, “thanks for that Jamie, I’ll see you around!”

Secretly, like Jamie, Sophie also wanted their conversation to prolong, but she didn’t really know what to say. After all, she was the princess clique type girl, and everybody knows that they don’t say much.

 Frequently asked questions What is instantaneous velocity?Velocity at any particular point in time. For example, the velocity after a certain amount of time, after it is thrown upwards, permitted to free fall, or thrown as a projectile.What is average velocity?Just as it sounds. The mean velocity.What are the uniform acceleration motion equations?Equations that can be used when acceleration is uniform (i.e. constant). They are a series of formulas, that link displacement, initial and final velocity, acceleration, and time.

Projectile motion is where there is not only a vertical component, but also a horizontal component present. Keep in mind that the components are independent of each other, and gravitational acceleration only acts on the vertical component, as it is a force downwards. As such, the motion can be separated into a vertical and horizontal component, and equations applied on to these components independent of each other. For example, acceleration of the vertical component is $10m/s^2$ down, but acceleration of the horizontal component is $0m/s^2$. The connecting factor is time. For example, a projectile shot at $100m/s$ at an $30^{\circ}$ angle above ground.

[img]projectile-motion-example.png[/img]

The vertical component (which will help determine time) is $sin(30^{\circ})=\dfrac{A}{100}$, meaning $A=100.sin(30^{\circ})=50m/s$. From this, we know that time in the air is $5s$ up, and $5s$ down, which is $10s$ in total time in air. The average velocity going from the start to the maximum height is $25m/s$, and the time taken to get from the start to the maximum height is $5s$. Therefore, the maximum height is $25\times 5=125m$. The horizontal component utilizes $cos$ instead of $sin$, and so is $100.cos(30^{\circ})=87$. The range, or the horizontal distance travelled, is $87\times 10=870m$.

 Frequently asked questions How does projectile motion differ from free falling?Free falling was just a drop (i.e. vertical movement). Projectile motion is where there's also a horizontal component present.How do we answer projectile motion questions?Remember that the vertical and horizontal components are independent.However, there are some things that are shared. For example, the time is shared. The vertical and horizontal components don't have different times. When the object starts, it starts. When it stops, it stops.What is range?The horizontal distance travelled.

Although mass and shape do not alter velocity per se, they do affect air resistance, which in turn alters velocity. Air resistance is a type of drag. Drag is a force which acts on an object in the opposite direction of velocity, due to air molecules. For the example of air resistance, the greater amount of air molecules hit, the greater the air resistance. Thus, shape (determining surface area) and velocity affect air resistance. Mass doesn’t affect the force of air resistance, but does affect the ability of air resistance to act against the momentum of the object.

 Frequently asked questions But don't mass and shape affect velocity?Not directly. They affect air resistance, which in turn affect velocity.What is air resistance?It's a type of drag. Drag is a force caused due to air molecules, opposing the direction of velocity.So what's exactly the effect of shape on air resistance?Increased shape increases air resistance.How about mass - how does that affect air resistance?Mass doesn't affect the force of air resistance. However, it does affect the ability of air resistance to oppose the momentum of the object.

 Formative learning activity Maps to RK1.6 What is a free falling body?

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