Ch3_ChungA

=toc Homework=

**__10/12/11 - Vectors Lesson 1 Parts A & B__**
Vectors and Direction:

Vectors have magnitude and direction while scalars only have magnitude



There are several naming conventions for vectors. Most use degrees in relation to north, south, east, and west.

The length of the vector corresponds to the magnitude.

Vector Addition:

You can add two vectors together to get a resultant

You can use the Pythagorean theorem to solve for the resultant of two vectors that create a right angle

You use trigonometry to solve for the angle (direction)

You can use the head to tail method to find the resultant of various vectors. This works well if you're doing a Physics "Walking Lab"

**__10/13/11 - Vectors Lesson 1 Parts C&D__**
Part C:

The resultant of several vectors is the displacement vector. The displacement of the resultant is the same as if you added up the vectors. (Holds true when you are talking about displacement vectors)

A vector doesn't have to be a displacement vector.

Resultant = sum of individual vectors

Part D:

Components make up the two parts of a two-dimensional vector

There is no difference between two forces that are pulling left and upwards and a force that pulls northwest. The same goes for any two dimensional vector.

**10/17/11 - Vectors Lesson 1 Part E**
Vector resolution is the process of determining the magnitude of a vector

Parallelogram Method: Draw a parallelogram around the vector, use a scale, and measure the X and Y components



Trigonometric Method: Draw a rectangle and use trigonometry to solve for the sides



10/18/2011 - Vectors Lesson 1 Parts F
You can add vectors using the addition method like this:



This uses the Pythagorean theorem and works when things are at right angles. However, you must change things up when things aren't at right angles.

You can re-arrange several vectors in order to get a resultant.



SOH CAH TOA and the Direction of Vectors:

You can simply get the angle of the triangles formed above using trigonometry. The tangent function works extremely well here.

Addition of Non-Perpendicular Vectors:

When things are non-perpendicular, you must use a different method to add the vectors. You can break down a vector into its components. For example, lets say we have a Vector X at 34 m and 23 degrees and a Vector Y at 50 m and 50 degrees. We would add them as such:

X's: 34cos23 + 50cos50 Y's: 34sin23 + 34sin23

Then, using the new added values, simply preform the Pythagorean Theorem to get the resultant and trigonometry to get the degree.

10/18/11 - Vectors Lesson 1 Parts G&H
Part G:

Often, things will move in a medium. A plane moves throughout air. As such, it's speedometer won't always represent how fast it is moving to somebody on the ground due to things like wind. This graph represents this idea:



You could also have side wind like this, in which case you would use the Pythagorean Theorem to find the resultant.



Riverboat problems are similar to airplane problems. You have to use the vector sum of the river velocity and the boat velocity.



Average Speed = Distance / Time

Part H:

The parts of a vector are independent and do not affect each other. All vectors can be thought of as having two components.



You can draw these components on a graph and then find the resultant.

Now, look at this:



The velocity components of this balloon are separate and independent of each other. The side wind doesn't affect how fast the balloon hits the ground. You can determine how fast it will hit the ground with the equation d = vt.

This is shown in the picture below:



The important thing to note is that affecting one WILL NOT AFFECT THE OTHER.

**10/19/2011 - Vectors Lesson 2 Parts A&B Method 3**

 * Lesson A: What is a projectile?**


 * Objective Statement: The only force acting on a projectile is gravity**


 * 1. What would happen to the motion of a cannonball shot from a cannon off of a cliff if there were no gravity?**

The cannonball would continue to move straight. It would not hit the bottom of the cliff because, since gravity doesn't exist, nothing would tie it down.


 * 2. Do you need force to keep an object in motion?**

NO! This question is one of the hardest to conceptualize. Force is only required for acceleration. Without force, an object would still have motion if not for gravity.


 * 3. What is inertia?**

Inertia is simply an object's resistance to changes in motion.


 * 4. How do we represent projectiles?**

Using free body diagrams. These diagrams also show the force of gravity, as gravity is the only force acting upon an object.


 * 5. What trajectory does an object in free fall take?**

An object in free fall, especially when shot out of a cannon at the top of a cliff, exhibits a free fall trajectory that could be best described as being parabolic in nature.


 * Lesson B: Characteristics of a Projectile's Trajectory**


 * Objective Statement: The components of a Projectile's Trajectory are independent of each other.**


 * 1. What are the two components of a Projectile's Motion?**

The two components of a Projectile's Motion would be horizontal and vertical.


 * 2. Does the vertical component of a Projectile affect the horizontal?**

No. The two perpendicular components of a Projectile do not interact with each other. One could say that they are independent of each other.


 * 3. Why do projectiles travel with a parabolic trajectory?**

Because the force of gravity directs the projectile into a parabolic trajectory.


 * 4. Does the velocity of a projectile's downward motion change?**

Yes, by a factor of -9.8 m/s/s (the acceleration due to gravity).


 * 5. Let's say that we shoot a cannon off of a cliff at an angle to the horizontal. Will it move horizontally or downwards.**

The projectile will move both horizontally and downwards. Since gravity doesn't affect the projectile's horizontal motion, it will still travel horizontally at the same time it is dropping. This creates motion in the form of a parabola.

10/20/11 - Lesson 2 Part C
1. Does the horizontal velocity of a cannonball shot off a cliff ever change?

No, it most certainly does not. It stays the same and is constant. The only thing that ever changes is the downward velocity (due to gravity). This is what causes the ball to eventually fall.

2. So, what is the horizontal acceleration of a projectile.

Zero. It doesn't change in speed.

3. What if an object is shot with an upward angle? Does the horizontal velocity change?

No. An upward angle simply means that we will have to use the Pythagorean Theorem to get the two components of the vector. The horizontal vector will stay the same while the vertical vector will change due to gravity.

4. What is the equation of an object's vertical displacement during free fall?

y = 1/2 * g * t^2

5. What is the equation of an object's horizontal displacement after being launched horizontally?

x = vi * t

=Labs=

Vector Class Assignment






Ball in Cup
The picture below shows the calculation of the initial velocity of our ball:



This calculation was used to find how far we should put the cup (theoretically):



Our actual cup was placed 3.04 meters away, rather than the hypothetical value of 3.09. Our calculated error was 1.62%, as shown by the calculation below.

[(3.09-3.04)/3.09] x 100 = 1.62%

There are several reasons for our percent error. The shooter isn't one hundred percent accurate. It is affected by things such as the angle at which it might be placed, where the ball is in the shooter, and the air in the room. Furthermore, the weight of the ball could vary if they got mixed up with another group. With all these various factors going into the lab, it makes sense that there would be some percent error. This project requires using a shooter that isn't even close to being perfect. Furthermore, there is a lot of human interaction with the shooter which makes it harder for it to be accurate. These things all led to the percent error of 1.62%.

Guardorama Project
Lab Partner: Sammy Caspert




 * Calculations:**

Distance: 1.7 Meters Weight: 2.1 kg




 * How Would I Change It?**

There are several things that could change about our project. First, our axles were rather weak and weren't secured properly. This affected the rate at which our wheels turned and subsequently how far our project was able to go. Second, we used an extremely weak base. We did this in an effort to cut down on weight. However, the weight of the pumpkin weighed down on the cardboard and caused the cart to slow down and skid. In this case, it would definitely be more advantages to use a wood or strong Styrofoam base. This would allow us to go further and not damage the project with every run. Last, our bearings were very old and should have been replaced with newer ones.

Shoot Your Grade
with Danielle Bonnett, Caroline Braunstein, and Sarah Malley

Hypothesis: Using the calculations we learned in class, we will be able to shoot a projectile out of a shooter angled at **20 degrees** through a series of five hoops into a cup.

Procedure: First we calculated the vertical heights and then hung tape rolls from the ceiling using string. We then shot the projectile out of the shooter through these five hoops.

Calculations:

The below picture calculates the initial velocity, a vital component to the entire lab.



The calculations below calculate where to put the cup.



The calculations below calculate where to put the rings, relative to the ground, so that the projectile will fly through them without hitting any.



The following calculations calculate the percent error of our lab.



Video of projectile going through 4 hoops:

media type="file" key="project.mov" width="300" height="300"

Conclusion: We were originally given the task of attempting to shoot a ball out of a shooter, at a 20 degree angle, through a series of five hoops. This seemed at first to be a daunting task that challenged our skills in mathematics and physics. However, I am proud to say that my group and I met the challenge readily. We calculated the initial velocity using the equations that we learned in class. We were then able to calculate the distances that the hoops needed to be from the ceiling in order to get the ball through them. This was a time-consuming process but the mathematics all checked out and we were able to achieve our goal. There were a lot of sources of error for this particular lab. For instance, the shooter wasn't always set at a correct angle. Often times the thing securing the shooter would move and the angle would change. Furthermore, balls were often mixed up which changed the trajectory of the shooter. The hoops often moved as a result of the fact that the strings would change over time and cause the hoops to either stoop downward or move to the side. Furthermore, since many groups were using our projectile we had to deal with the variables that came with that. If I were re-doing this lab, there would be several changes that I would make. First, I would not allow other groups to use our project. I would also invest in another projectile that is more accurate and steady. I would also separate the groups from being so close to each other as balls were often lost or mixed up. All in all, I would say that our project was successful. Our percent errors were very small with the greatest one being 1.47%. Our calculations weren't always 100% accurate as the percent error shows. However, a little bit of physical manipulation went a long way and we were able to get the ball through the hoops, as our video shows. This project was incredibly gratifying and my group and I gained a lot of pleasure from watching the ball fly through the hoops as per our calculations.

=Notes=


 * 10/25/11**

Projectile motion at an angle:

D = ViT + .5at^2 0 = 10sin25 * t + .5(-9.8)t^2 t = 0.88s
 * || x || y ||
 * Vi || 10cos25 || 10sin25 ||
 * a || 0 || -9.8 ||
 * t ||  ||   ||
 * d || range? || 0 ||

Dx = 7.97

How do we find the time to maximum height?

tTotal /2 = tMax .88/2 = .44

How do we find the maximum height?

Use the Dy = ViT + 1/2at^2 equation and plug in .44, which is the time to maximum height