3.2 Acceleration Vector

Acceleration describes how a velocity of a particle changes over time. So our equation is very similar to the velocity vector equation. It is (V2-V1)/(time2-time1) basically the change in velocity over the change in time.

Now for those of you who know calculus, instantaneous acceleration at any point is the limit of the average acceleration vector when x2 approaches x.

The acceleration is tangent to the path on if the particle moves in a straight line.

This is an example from University Physics text book, "to convince you that a particle has a nonzero acceleration when moving on a curved path with a constant speed, think of your sensations when you ride in a car. When the car accelerates, you tend to move inside the car in a direction opposite to the car's acceleration. Thus you tend to slide toward the back of the car when it accelerates forward and toward the front of the car when it accelerates backward. If the car makes a turn on a level road, you tend to slide toward the outside of the turn; hence the car has an acceleration toward the inside of the turn."

There is always an equal and opposite reaction.

Having said that, lets do a practice problem from University Physics with acceleration:

A jet plane is flying at a constant altitude. At time t1 = 0 it has components of velocity Vx = 90m/s, Vy = 110m/s. At time t2 = 30seconds the components are Vx = -170m/s, Vy = 40m/s. a) Sketch the velocity vectors at t1 and t2. How do these two vectors differ? For this time interval calculate (b) the components of the average acceleration, and (c) the magnitude and direction of the average acceleration.

Will post answers 1/17

Mark Jackson
Physics/Political Science Student
Metropolitan State University of Denver

3.1 Position and Velocity Vectors

Answers to the review questions for the end of Chapter 2. To enlarge the image, all you need to do is click on it.

So I have been out of the game for a little while due to some school work that needed to be done. But now I am back and ready to work.

We are now onto another basic yet fundamental function of physics. Position, velocity, and acceleration vectors.

Position vectors relate to a particle at a point in time (pretty basic huh?)

Velocity vectors have to do with a particle moving toward one position to another over a change in time. Basically it is the change in position over a change in time.

However velocity has a little twist to it. This is twist is called instantaneous velocity. Instantaneous velocity represents the limit of the average velocity as it approaches zero in relation to the change in position in time.

I will post the next set of lecture slides late today on acceleration vectors.



Mark Jackson
Physics/Political Science student
Metropolitan State University of Denver

Answers to 2.5; review questions for ch. 2

So here are my answers from the previous post. I made these answers at midnight last night. If your cannot read any of the numbers or words please comment.

And now here are the next questions. Courtesy of University Physics

1) You normally drive on the freeway between San Diego and Los Angeles at an average speed of 105 km>h 65 mi>h, and the trip takes 2 h and 20 min. On a Friday afternoon, however, heavy traffic slows you down and you drive the same distance at an average speed of only 70 km>h 43 mi>h. How much longer does the trip take? 

2) The fastest measured pitched baseball left the pitcher’s hand at a speed of 45.0 m>s. If the pitcher was in contact with the ball over a distance of 1.50 m and produced constant acceleration, (a) what acceleration did he give the ball, and (b) how much time did it take him to pitch it?

3) You throw a glob of putty straight up toward the ceiling, which is 3.60 m above the point where the putty leaves your hand. The initial speed of the putty as it leaves your hand is 9.50 m>s. (a) What is the speed of the putty just before it strikes the ceiling? (b) How much time from when it leaves your hand does it take the putty to reach the ceiling?

  

2.5 Free Falling Bodies

There is an acceleration that we deal with every single day. Gravity. We will use this for our everyday practices in the mechanics portion of physics. So this section we will discuss various problems and then revisit them either tomorrow (7/26) or (7/27). This will be the last section before the end of chapter 2. We will begin chapter 3, for sure, on Sunday July 28th.

1) A one-euro coin is dropped from the Leaning Tower of Pisa and falls freely from rest. What are its position and velocity after 1.0s, 2.0s, and 4.0s?

2) You throw a ball vertically upward from the roof of a tall building. The ball leaves your hand at a point even with the roof railing with an upward speed of 15.0 m/s; the ball is then in free fall. On its way back down, it just misses the railing. Find (a) the ball’s position and velocity 1.00 s and 4.00 s after leaving your hand; (b) the ball’s velocity when it is 5.00 m above the railing; (c) the maximum height reached; (d) the ball’s acceleration when it is at its maximum height.

Use the equations we learned in the last section. Again we will revisit these problems on 7/27

Mark Jackson
Political Science/Physics Student
Metropolitan State University of Denver

2.4 Constant Acceleration

First off constant acceleration means, as it states, your acceleration will stay the same no matter what. However,  does this mean that our velocity will stay the same too? No. In fact, our velocity will be increasing, either in the positive direction or in the negative direction.

So the equation for constant x-acceleration we should know is:
x-velocity = (x-velocity at time = 0) + (the product of x-acceleration and the time interval)

I do not have any equation tools on Blogger so we will have to stick to this.

Another equation is when the constant acceleration has a x-velocity that changes at a constant rate.
This equation will be expressed as

average x-velocity = (initial velocity + later time velocity)/ (2)

However this equation will fail if the velocity changes over time. The equation we must express is:
average x-velocity = initial velocity + .5(the product of x-acceleration and the time interval)

Seems pretty straight forward but anyone can screw this up quick. Pay attention to details. Now we can combine all of these equations to form one equation..

x = initial position + (the product of initial velocity and a later  time) + ((.5) (the product of acceleration and time ^2)

Here are some other equations that relate to constant acceleration. Each equation has something different about it. One may include time and not include acceleration. You must distinguish between the two when doing problems.

V_x² = V_0x² +2a_x(x-x_o)

x-x_o = ((V_ox + V_x ) / (2))

Average and Instantaneous Acceleration 2.3 -- Answer to Question 2.2

So first off, here is our answer to the question from the previous section -------------------------------------->>

We will likely have another question as soon as we finish 2.4 which will likely be tomorrow the 24th.

Now onto Average and Instantaneous Acceleration. Similar to instantaneous velocity, but not quite.

Instead of using the distance between the two objects, average velocity, we use the velocities between two time intervals. It is basically the change in velocities over the change in time.

Now instantaneous velocity is also similar to instantaneous acceleration. Instantaneous acceleration is the derivative of velocity with the respect to time.

I was going to post a fantastic graph but I cannot get it to copy and past correctly onto the is blog. So I will explain it.

The greater the curvature (upward or downward) of an object’s x-t graph, the greater is the object’s acceleration in the positive or negative x-direction.

Hopefully that makes some sense. Honestly the graphing part is not as important as the mathematical part of physics. Once you understand how the graphs work, you will understand the majority of graphs you will look at through physics.

Next topic will be on constant acceleration and free falling bodies. BOTH important topics

Mark Jackson

Instantaneous Velocity 2.2

In certain cases, as physicists, we want to solve a problem that deals with a speed at a certain time in space. This is instantaneous velocity. Remember this is only along a straight path. Later we will deal with more complex examples.

As the change in x and the change in t(time) become extremely close to each other. In terms of calculus, we would say the limit of delta x over delta t is our derivative x with respect to t, which is written at dx/dt. 

The time interval, we must remember, is always positive. Instantaneous velocity is a vector quantity. One other thing a physics student should know is the difference between speed and instantaneous velocity. Speed measures how fast something is moving. Instantaneous velocity is the measure of how fast and what direction a particle is moving.

Another part of instantaneous speed is that it does not matter whether or not the velocity of something is negative or positive. For example, lets say a particle is moving at a rate of -67 m/s and another is moving at 67 m/s. The instantaneous speed is still 67 m/s

Lets do a problem from the book University Physics

A cheetah is crouched 20 m to the east of an observer . At time t = 0 the cheetah begins to run due east toward an antelope that is 50 m to the east of the observer. During the first 2.0 s of the attack, the cheetah’s coordinate x varies with time according to the equation x = 20 m + 15.0 m>s22t2. (a) Find the cheetah’s displacement between t1 = 1.0 s and t2 = 2.0 s. (b) Find its average velocity during that interval. (c) Find its instantaneous velocity at t1 = 1.0 s by taking ¢t = 0.1 s, then 0.01 s, then 0.001 s. (d) Derive an expression for the cheetah’s instantaneous velocity as a function of time, and use it to find vx at t = 1.0 s and t = 2.0 s. 

We will revisit this problem tomorrow(7/23)

Mark Jackson

Answer Ch.1 --- Chapter 2.1

So here was our question:

1) Hearing rattles from a snake, you make two rapid displacements of magnitude 1.8m and 2.4m. In sketches roughly to scale, show how your two displacements might add up to give a resultant of magnitude (a) 4.2m (b) 0.6m (c) 3.0m. 

And here is our answer:

Now on to Chapter 2

So the first and one of the most integral parts of physics is mechanics. The simplest form of physics is velocity and acceleration. Previously we learned about velocity through the study of vectors. 

Now acceleration and velocity of a very unique relationship. They are derivatives and integrals of each other. But before that we must know how to calculate displacement, time, and average velocity. 

A displacement is the distance or where a vector points to which are labeled as x1 and x2. So lets say we have an airplane sitting on a runway. Position 1 starts at 30 m and 0 seconds. Now lets say an airplane travels 1000 m down the runway and ends up taking the airplane 15 seconds to reach that take off point. What is our m/s?

So we take distance/time. distance is (1000m - 30m)/ our time (15s - 0s). This answer gives us 64.67 m/s. This equates to our average velocity. A very simple topic to grasp when relating to straight line motion. 

Beginning July 18th we will go over the topic of instantaneous velocity. Calculus is involved. 

Mark Jackson
Metropolitan State University of Denver

Vectors 1.3 Part 2

Unit vectors are important in physics but not important yet in our stage of learning in physics. So if you want some really good information, go to http://www.youtube.com/watch?v=lQn7fksaDq0. Khan Academy provides this information. Listen if you have any questions that I have not covered please go to this Youtube site or his site khanacademy.com.

Now we have reached the end of chapter 1. This is our first chapter of our venture in physics. But before we move onto Chapter 2, we are going to try and solve a problem dealing with vectors. I will provide the answer tomorrow. Again please email me or go to khanacademy.com for further questions and answers. 

Question (Courtesy of University Physics)

Vector and Vector Addition

1) Hearing rattles from a snake, you make two rapid displacements of magnitude 1.8m and 2.4m. In sketches roughly to scale, show how your two displacements might add up to give a resultant of magnitude (a) 4.2m (b) 0.6m (c) 3.0m

ANSWER TOMORROW (7/16)

Mark Jackson

Metropolitan State University of Denver

Vectors 1.3 Part 1

Vectors may be hard to understand to many physics students. Personally, it took me about a month to finally have a grasp on what exactly a vector was or did.

First a scaler quantity has a measure of just magnitude. A vector quantity has both magnitude and direction. A scaler quantity is just 3m/s x 3m/s = 9m/s. Vectors use a different sort of symbols and how they are measured.

Lets say we have a car moving at 30 m/s at 0 degrees north. The vector would point to the top of the page and the symbol for the vector would be an A with an arrow over the A. Now when you have two vectors pointed the same and have the same magnitude this referred to as "Parallel Vectors." Now when we have two vectors moving opposite directions, this is called negative vectors. The symbol for negative vectors is A=-B or -B=A. We refer to this as antiparallel.

We must remember the the magnitude of a vector quantity is always positive. Drawing vectors is like drawing a scale for a map. Lets say 1 centimeter drawn equals 9 km and 2 cm's equals 18 km. So you must draw a scale when you draw your vector.

Vector Addition and Subtraction
Another simple but tough area of physics to grasp. Remember this equation C=B+A. This looks like Pythagorean's theorem doesn't it? Well this is somewhat similar. C is the hypotenuse of our equation and B+A are the arrows that add for C. However, this does not at all mean the magnitudes are the same. The magnitudes depend on the angle of the two vectors and the magnitudes of those two vectors. Also if the vectors are opposites of each other then we subtract the two vectors.

Next lesson will be tomorrow July 10th

Questions email me at mjacks66@msudenver.edu

Mark Jackson

Uncertainty and Sig Figs 1.2

This is probably the most boring topic a physics student will ever study in his or her career. Uncertainty of measurements and significant figures. But they are very important when it pertains to physics.

Uncertainty is a difference in measurements between two different devices. Lets say we have a meter stick and a millimeter caliper. We are going to use these to measure the width of a novel book. The meter stick will have a greater uncertainty than the millimeter caliper. So using the right measure instrument for the job is important.

Significant figures are also important, but boring, when it comes to physics. Again, lets say you want to measure the speed of an electron moving about on an atom. Would you want to measure that in Km/Hr, absolutely not. An electron moves extremely fast. Theoretically, lets say, an electron is moving at the speed of light. The speed of light is 300000000 meters/second. About. So how many sig figs is that number? 1. We must turn that large number into scientific notation. 3.0E8. This is the right way to express the speed of light. But I think I forgot something. Can you figure it out?

Uncertainty and sig figs are important aspects of physics. You will use them in everyday physics. Not just in your boring lecture halls.

P.S. I forgot my units. 3.0E8 m/s

Questions? Email me at mjacks66@msudenver.edu

Mark Jackson

Intro to Measure 1.1

Time and length are the most common units of measure. Since the early eras, cultures have used the sun for their time. But in our modern era time is more precise. How does time become so precise? Scientist use the atomic clock. The atomic clock uses a precise microwave that measures the energy difference between two cesium atoms. One second in measured to about 9 billion cycles of microwave radiation on that atom. There's something you don't learn everyday.

A common problem for many students, I included, struggle with remembering to insert units to our answers.

For example, lets say you put 1 as your answer. 1 what? 1 apple? 1 water bottle? A physics student must include his or her units. This is imperative. You will get marked off if you don't include units.

Units must be dimensionally consistent. By that I mean you cannot add buildings and broccoli together. If you have a time and a distance you can measure that, by the equation d=vt. Or a speed and a time which gives you distance.

Let's do some practice units.
How many grams are in a kilogram?
How many nanoseconds are in on second?
Answers further down on the page.

There are two different standards of measure. The SI and the US measure system. For our purposes we will be using the SI. Kilogram, meter, and liter are all examples of the International System.

There are 1,000 grams in one kilogram
There are 10^9 nanoseconds in one second

Questions
Email me at mjacks66@msudenver.edu

Mark Jackson
Student Metropolitan State University of Denver

Teach yourself physics too!

Welcome all!

So I will be teaching myself all the levels of physics over the next five years. From intro to motion, electrostatics, through the theory of relativity and quantum mechanics.

This will be tough. But if you just keep your head in it, anything is possible.

As we move along through physics, you will need some understanding of Calculus and differential equations. But Diffeq will not be until the end of the second year, probably the third year.

Every two weeks or so I will be posting a problem online, then a day later I will post a solution on how to solve that problem. But the most important concepts of that chapter will be introduced.

So the first chapter we will be studying is Introduction to Motion. But I will have to leave ya'll hanging here because it is getting late and I need rest for my day of teaching physics.

I will post my first chapter problem and lecture on Monday July 8th.

Questions call me at 720-984-8777 or email me at mjacks66@msudenver.edu

Thanks!

Mark Jackson
Physics wanna be teacher and enthusiast
Student at Metropolitan State University of Denver