CAPM Model Summary

     This unit was the most elaborate one this year to me. I had the opportunity to learn many new concepts, while going back and visiting older ones from earlier this year. The CAPM model, or Constant Acceleration Particle Model, really showed and explained exactly what we mean as physicists when we want to describe speed, or acceleration. In this quest for knowledge, I connected many different parts of previous knowledge to form what I now know. The following is a brief summary of my understanding of the constant acceleration model as of now.
     The first thing that I learned was the idea of instantaneous velocity of an object, or the velocity at a certain point in time which an object is traveling at. I found that there is actually two different ways to find the instantaneous velocity of an object: by finding the slope of the tangent on a position versus time graph, or by using the mathematical equation V(final) = acceleration*time + Velocity (initial). Finding the slope of the tangent seems way harder than it actually is; all that's needed is at least three different points on a position versus time graph. Here's an example: I want to find the instantaneous velocity of an object at 6 seconds into the test. So, I'll use points 5 and 7 to test...

V(final)= a*t + V(initial)

 V= (change in)X/ (change in) T

122.5 m- 62.5 m / 7 secs - 5 secs =

60 m / 2 secs =

30 meters / sec

That's one way to find the instantaneous velocity of the object. However, if I didn't have a position versus time graph to use, I can always use the mathematical formula V(final)= a*t + V(initial).

This requires me to be using a velocity versus time graph, and simply finding the slope of the graph. I'm able to do this because the slope is also my acceleration.

     The next thing that I learned was how to find the displacement or distance an object travels in a given amount of time by using a velocity versus time graph and by using the model x = 1/2 a*t + v. I already know how to find the displacement of an object using other types of graphs, but this gives me a quicker, more efficient way to an answer. When I use the velocity versus time graph, I have to find the area under the curve, which amounts to the displacement. Here's an example problem:
Imagine a invisible triangle connecting the line to the sides of the graph. This space should look like a triangle and a rectangle. This is what you need to find for the displacement.

15 m/s * 6 secs/ 2 = 45 m/s^2+

15 m/s * 6 secs = 90 m/s^2

Displacement = 135 m/s^2 



Now, to the heart of this unit; acceleration! I actually have four different ways to find the acceleration of an object, all of them listed out below:
- Finding the slope of a Velocity versus Time graph
- Using the model a = (change in) Velocity/ (change in) Time
- Using 1/2 a * t + Velocity (initial)
- Using Velocity (final)= a * t + Velocity (initial)
I've already showed an example problem involving three of these four solutions, but I haven't touched on when to use 1/2 a *t + Velocity (initial). I should only use this equation whenever I have a Position versus Time Squared graph and need to use it to find the acceleration. However, I can only do this whenever the starting velocity is equal to 0. The trick to using this method is to not forget to remember that the slope is only equal to 1/2 of the acceleration, so I need to always add back another half to find my full acceleration.
   
      A big part of my first unit at the beginning of the year was learning all about graphs. More specifically, position versus time and velocity versus time. In this unit, I expanded on this previous knowledge by adding acceleration versus time graphs. I also found how to graphically represent an object when the surface isn't flat. This in turn changes my position versus time graph from having a straight line with a constant slope to either an exponential growth or decay graph. I developed new language to describe these new graphic ideas. Despite adding all of these new ideas, I can still find things such as the displacement the same way as before; by finding the difference in the distance from the starting point to the ending point.
   
     The same thing applies to my previous knowledge about velocity versus time graphs. I still retained all the same principles and ideas from the first unit, I just added some new knowledge to the mix. However, this time I played with the idea of a velocity versus time graph having a constant slope, meaning that the acceleration (as the slope) is staying constant as well. This meant that my velocity graph was no longer just a straight line; it began to look more like a position versus time graph from the first unit. This took on a different meaning, depending on which way the object is traveling and if it's accelerating or decelerating. Here's an example graph below:


We can tell that the object is accelerating because it's covering more distance in less time as the test runs. However, this acceleration is towards the sensor, because the position from the sensor is getting smaller. My velocity versus time graph would start down below the line, as the object is moving towards the sensor, but be moving from the origin down at a constant pace. The velocity graph is constant because the object's speed is constantly increasing, which is the acceleration (also the slope for velocity versus time). This summary has giving me a clearer picture of what I have learned going into the test, and I hope to revisit this throughout the year for additional help if needed!


   

BFPM Challenge Problem

This is a diagram of our setup, with  the mystery weight attached to the ceiling by two points on the ceiling. It might be hard to tell, but the angles of each line are not the same. 

This is what my setup looked like when broken down into a free body diagram with every force acting on the object. I broke the two tension forces up because every force has to be on the axis of the  diagram to be equal to another force. This diagram shows that it is balanced, as every force is accounted for on the axis. 

Once I had my free body diagram, I could begin to find my predicted weight, or the Force of Gravity that is acting upon the mystery weight. I first measured the angles at which the lines were holding the weight, and they showed Line A to have an angle of 35 and Line B to have an angle of 70. With this information, I could then use Soh Cah Toa to find the exact amount of force in Newtons that FTenAy and FTenBy have. I'll start with FTenBy. Since 20+70 is equal to 90 degrees, I can use the cosine of 20 to find the amount of force in FTenBy. My math is as follows:

Cosine(20)= FTenBy/ 2.2 (N)

FTenBy= Cosine(20)(2.2 N)

FTenBy= 2.1 N

To solve for the amount of force in Newtons for FTenAy, I chose to use sin(35) to solve, because sin is equal to opposite/hypotenuse. The following is my setup and solved answer:

Sin(35)= FTenAy/0.8 (N)

FTenAy= Sin(35)(0.8 N)

FTenAy= 0.46 N

Even after all that work, I'm still not entirely finished. The two force amounts that I found still need to be added together, as both of them come together to equal the force of gravity on an object, or more commonly known as the object's weight. 

FTenBy+FTenAy=Fgravity

2.10 N + 0.46 N= Fgravity 

Fgravity= 2.56 N

CAPM Challenge Problem

This is a sketch of how our setup looks like. The green objects represent the books under the table, with the table being the black line. The orange object is our metal ball, which rolls faster when rolling down the table (look for the red arrows). We measure our spacing in centimeters, timed to half-seconds for the most accurate measurements. 

     

0	0
0.5	5
1	16
1.5	32.5
2	54
2.5	80
3	110
3.5	144

Our raw data looks like this, with the x-coordinate being the half-seconds we measured and the y-coordinate being the distance we measured in centimeters.

Our first graph shows our raw data in a position versus time graph, proving our guys correct that our object would move with an exponentially increasing speed. We were originally going to use this data and create a Velocity vs. Time then find the slope of that line. That would then give us the acceleration of our ball. However, we could easily cut down on the number of steps to do by instead using a Position vs. Time Squared graph.

This is the result of squaring our time and using the new results instead of the old time, which turns our exponential data graph into a straight line that has a slope. The equation for this line is equal to X=1/2 at^2, with the slope of this line being equal to a half of the acceleration of the object. Our slope, as seen in the graph above, is equal to 11.708 cm, or .11708 meters. we took our new slope and multiplied it by 1/2, which turned out to look like this: .11708/1*2/1. This all resulted with our acceleration being equal to .23416 meters/second squared. 

Unit Blog Summary #2

     Well, I've certainly covered a bunch of material so far, and this section has been the most rewarding so far in terms of my overall learning and growth. I went from barely understanding which forces act on an object to knowing the precise amount of force in newton’s (N). In the seven sections that I covered in class, everything built on top of the last section, making it extremely important to retain my prior knowledge. The first thing we learned was how to draw a free-body diagram showing forces acting upon an object. These forces that we learned first were gravity, normal, and friction forces. Also, by using proportionately sized force vectors (Lines used to measure the amount of a certain force), I was able to show if opposite forces were equal or different. 

     Moving forwards a couple of days and some learning opportunities, we learned about Newton's first law. It states that an object in motion will stay in motion, and an object at rest will stay at rest, until another force acts upon that object. For example, if I had a ball rolling across a frictionless floor at a constant speed, the ball wouldn't stop unless another external force is applied to the ball.

However, as soon as I apply a force such as friction, the ball would slow down and eventually stop moving all together.  A tricky example of this is a ball that I've just released from my hand upwards. The only force acting on the ball throughout it's entire flight is gravity, as it's slowing down before reaching the vertex, changing direction at the top of it's arch, and then speeding up as it travels downwards. This unit gave me the tools to realize how individual forces act upon an object.

     The third major thing I learned this unit was to realize the difference between the mass and weight of an object. The mass is the measurement of how much material an object has, while the weight is the measure of how hard gravity acts upon an object. The units for the mass are in kilograms (kg) while the weight is measured in newton’s (N). To convert from mass to weight, you need the following equation: Weight= (10)*(mass). Note that the 10 stands for Earth's gravity constant that’s actually 9.81, but 10 is okay for most problems. The mass also has to be in kilograms. One cool thing that I learned was that the only force acting on an object while it was in the air is gravity, nothing else. With critical thinking, this begins to make perfect sense, as it is the only thing acting on an object while it's slowing down, speeding up, or changing directions in the air.  

     The next thing that I learned was how to find the actual amount of friction that acts on an object. To find this, you need the weight of an object and the coefficient of friction, which is measured between the two surfaces of the interacting objects. The actual equation looks like this: F=(mx)*(weight), with mx as the coefficient of friction. An important idea that I learned was the notion that friction is not affected by the surface area of an object. The only two things that do in fact affect friction is the weight of the object and the speed that it’s traveling. Also, it's important to note that the direction of the velocity of an object is directly related to the direction of the unbalanced force in a free-body diagram. If an object has an unbalanced force going the same direction as velocity is, then the object is speeding up. The opposite occurs if they are facing opposite directions. 

            One of the most recent things I learned was to realize when to shift my axis, like in the many

ramp problems I solved. After that, I moved to Newton’s third law, which states that each action has a equal and opposite reaction. This was helpful in discovering which forces between different force diagrams act on each other, such as in the tug of war game between Asheville School and Christ School. It’s also very important to remember and draw the force vectors between the two diagrams as proportionate to one another.

            The most recent thing that I discovered was how to plug in values for each of the forces I previously had. More specifically, I looked at the force of tension and using sine, cosine, and tangent to find the specific amounts of force in newtons along lines. Using these three different equations also allowed me to find various force amounts on other forces. I also got the chance to pull together old concepts from earlier this year and add to them. For both the horizontal and vertical line equations, I can plug in numbers instead of force names and use them to check my answers. For example, if I said that the Force of Tension+ the Force of Friction= 0N, then I could take my numbers for the forces and check them to make sure they match up with everything else. Altogether, I finished up this unit with a strong understanding of how objects can be broken down into their diagrams a lot easier than I originally thought. In the BFPM challenge problem, I got a real life opportunity to apply what I learned to solve for the weight of an object. I felt very well prepared for all of my assignments, and I’m looking forward to crushing this test!

Texting While Driving Challenge Problem

Texting While Driving

By converting our average drive speed into meters per second, we find one part of the formula V= the change of distance/ the change in time, the average velocity of the car. By doing this, we decided to use this formula to find the total distance travelled. We then switched the variable to solve for the distance and plugged in our known info (velocity, time) to find our distance travelled over the span of time.

65 miles/1 hour*1609.34 meters/1 mile*1 hour/60 min*1min/60 sec= 29.06 m/s

Takes 1.4 seconds to text LOL

*plug this in* 

x=29.06 (1.4) 

x= 40.684 meters travelled

Unit Summary Blog Post #1

Unit Summary Post

     In this unit, I got only a taste of the world that is physics. We touched on several things, starting with how to put data into Excel and create equations with lines. If the data is curved exponentially, I can plug in various equations to help straighten my data and make it reliable to make predictions from. One big thing that I learned in this unit was how to determine if a set of data is reliable or not. For example, when plugging in data points into Excel, you can use the coefficient of determination, or r^2, to see if your data has a strong connection or not. The closer the coefficient is to 1, the closer the points will be to the line of best fit.  From this reasoning, I can work out either to use the data or find a different reference point using some other connection between the data points. From there, we moved into learning how to apply these graphs in real life situations; one big thing I learned out of this section was how to determine which direction an object is traveling depending on if the slope is positive or negative.

      A big lesson that I learned was in our first big group project with the rubber bands and blocks of wood. During the lab, our group forgot to take data during a crucial part of our testing. Because of this, our data was warped and we couldn't use it. By learning to always take care and record everything you find, I know to be meticulous in my findings and always prepared for variables during testing. Unfortunately, our sketch of our lab that shows what was needed for our lab isn't available, but we had a rubber band stretched across two holders that shot a wooden block a certain distance depending on how far you pulled back on the band.

     Learning how to carefully track my progress came in handy when I had to label different x and y axis on graphs. Another thing that I learned so far this year is how to find the velocity of a object. Velocity is defined as the change in distance over time, or V= (change of) x/t. We used position versus time graphs and velocity versus time graphs to measure and track certain objects, then used that information to describe the said object. I got the chance to apply these skills during the buggy lab, which had us find our cart's average velocity. Once we had found that, the challenge went up by a bit. We partnered with another group and had to determine the exact location at which they would crash if we set them up a certain distance away from each other. The equation we used was Position=(Velocity)(Time)+The starting position. I found that by setting each equation equal to each other, we could find the x-cordinate of the point of interception. By plugging that point back into either equation, I found our predicted location of where our buggies would collide. This lab proved to be one of the toughest so far, but the reward at the end of the process made it worth the effort. I hope that the rest of this year will yield even greater rewards of knowledge for me.