Physics Challenge- Wind cars

In this lab, we used motion sensors to get two equations of different wind powered cars. These cars both accelerated from rest. In the lab, we tried to predict where the cars would have to start in order to intersect at the origin. Our work is as follows:

Constant Acceleration Particle Model

Position vs Time Graphs: Previously we have known that when a position vs time graph has a straight line on it, the object is moving at a constant velocity. With the CAPM model, we learned that these graphs can also have a curved line, which indicates acceleration. Also, we know that steeper slopes indicate that an object is moving fast, and that when an object moves in the positive direction it is going forwards and when it is going towards the negative side of the graph that it is going in the negative direction. We can also apply this information to our new curved lines.

As the line moves down as the vector
indicates, the velocity is increasing.
As the vector going up indicates, the
velocity is also increasing. 

If a velocity line is going up in the direction
 of the top vector, the object is moving in
the positive direction, and if the line is going
downward like the bottom vector, then the
object is moving in the negative direction. 

If the line moves toward the origin over
 time line the top vector does, the object
is slowing down. Also, if the line moves
towards the origin like the bottom vector,
the line is also slowing down. 

If the line on the graph moves towards
the origin like the top vector, the
object is still moving in the positive
direction. Similarly, on the bottom vector,
the object is moving in the negative
direction. 

In this example, the object starts with
a velocity of 10m/s. From seconds 0-2, the
object is slowing down. Then, as it crosses
the origin, it changes direction and
begins speeding up in the negative direction.

In this graph, the object is increasing in speed in the positive direction. To calculate displacement,
we can calculate the area between the line and the x-axis. This area is in the shape of a triangle, so
we can use Displacement = 1/2 (Base)(Height) to find displacement. By substituting values in from
the triangle that the line makes with the x-axis, we create an equation that should look like this:
1/2(10)(10) = displacement. From there, we can conclude that the object finished 50m from where it started.
To take a different approach, we can also use the equation ΔX = 1/2aT^{2 + initial velocity} to find the displacement. Initial velocity is the velocity the velocity at zero seconds. In this case, initial velocity is zero. Also, we can find the acceleration by using the equation (ΔV)/(Δt) = a. We can only use this equation for acceleration when our velocity vs time graphs are a straight time, indicating constant acceleration. In this case, the acceleration is 1m/s^2. After we find acceleration, we can plug in our numbers and solve for the displacement. ΔX = 1/2(1)(100) + 0 which comes out to equal 50m, the same as our first calculation.

Average Acceleration:

Formula: 

Average acceleration is the average rate at which an object is speeding up. Average acceleration may be very useful for many other equations and formulas in physics. 

In this graph, to find the instantaneous velocity at five seconds we can
find the slope between two pints that will make 5s the midpoint.
 For example, we can find the slope between the points at 4s and 6s.
This line is also tangent at five seconds. The fourth second falls at 1m and
the sixth at 2m. Therefore, we can use the equation (change in position)/(change in time) 
to find the line tangent. This tangent line will be the instantaneous velocity at that point. 

Example: 

In the position vs time graph above, the object has four distinct motions. From point A to B, the object is moving in the positive direction. Also, since the line is curved, we can determine that it is either slowing down or speeding up. The way to differentiate between the two is look at the slope of the line. At the beginning of the line, the slope is very steep. It gradually becomes less steep until the slope becomes zero at point B. With this information, we can tell that the object is initially going very fast and then slows down. From point B to C, the object is moving towards the negative side of the graph, telling us that the object is moving backwards. The slope closest to point B is not very steep, but gets steeper as it gets closer to point C, indicating that the object is speeding up. At point C something interesting happens. The object crosses the origin with a very steep slope (high velocity) and then begins to slow down again. This can be seen with the gradual decline of slope. At point D, the object switches direction again and begins to travel in the positive direction again. Starting with a small slope, the object begins traveling slowly and then speeds up until it reaches the origin, finishing at the same position  that it started at. 

Velocity vs Time Graphs: Constant velocities are shown on a velocity vs time graph as a horizontal line. When a velocity vs time graph has a diagonal line, the velocity is either increasing or decreasing. An object with an increasing or decreasing speed has an acceleration. As seen on the velocity vs time graphs below, as the line moves away from the origin, the object is increasing in speed. Also, if the line is going in the positive side of the graph, it is moving forwards (above the origin) and if it is in the negative side of the graph, it is moving backwards (under the origin) 

Example: 

 Displacement and Velocity vs  Time graphs: We can use two methods to find the displacement from a velocity vs time graph. The first is to find the area of the space between the line and the x-axis. This space represents how far the object has traveled. Also, we can use the equation ΔX = 1/2aT^2 where ΔX = displacement, a = acceleration, and t = time. 

Example problem: 

Acceleration vs Time Graphs: Acceleration vs Time graphs do not tell us much other than the slope of the velocity vs time graph. a vs t graphs are directly related to the slope of v vs t graphs. For example, if the slope of a velocity vs graph is negative, the acceleration line will also be on the negative side of the graph. Keep in mind that there is no acceleration when the velocity vs time graph has a straight horizontal line with no slope. 

Examples: 

In this velocity vs time graph, the line has a positive slope from 0-5 seconds, no slope from 5-15 seconds, and then a negative slope from 15-25 seconds. Because of this, we can conclude that the acceleration for the first five seconds is positive, there is no acceleration for the next ten seconds, and then there is a negative acceleration for the last ten seconds. The acceleration vs time graph would look as follows: 

Motion Maps: We have now learned how to make both acceleration and velocity motion maps. In velocity motion maps, the distance between the points represents how far the object has traveled and the vector represents the velocity. The larger the vector, the larger the velocity. In acceleration motion maps, the distance between dots represents distance traveled, but the vectors represent acceleration. Constant acceleration is represented by vectors of the came length as seen below. In the image below, the object's velocity is increasing with constant velocity. 

Average Velocity: Average velocity is calculated by the equation (change in position)/(change in time) and represents the average velocity that an object travels at over a given amount of time. 

Instantaneous Velocity: Instantaneous velocity is the velocity at a given time. To find instantaneous velocity on a position vs time graph, you can find a line tangent to the point at which that time is located (also the same as making that time the midpoint of two other points on the line). 

Real World Examples: Physics is ever present in the real world. One example of a real world situation where acceleration is present is as follows: A dog runs down his driveway with an initial speed of 5m/s for 8s and then uniformly increases his speed to 10m/s in the next five seconds. How long is his driveway? 

How to solve: So, we are given both time and velocity. Since we are trying to solve for displacement, we can use the equations ΔX = 1/2aT^{2 + initial velocity} and x = Vt. In the first eight seconds of the run, the dog has a constant velocity. Because of this, we use the exertion ΔX = Vt. After substitution, this becomes ΔX = (5m/s)(8s), simplified to 40m. Keep in mind that this is only the first half of the dog's journey. The second half can be calculated with the equation ΔX = 1/2aT^{2 + (initial velocity)(time)}

. We can find acceleration with the equation ΔV/Δt which comes out to be 1m/s^2. After substitution, this equation becomes ΔX = 1/2(5)^2 + (5)(5) This can be simplified to ΔX = 37.5. So, now that we have both the first part of the journey and the second, we can add them together in order to find the total displacement (length of the driveway). 40m + 37.5m = 77.5m, the length of the driveway. 

BFPM Practicum

Balanced Force Particle Model

Introduction:

      In Unit two of our physics class, titled "Balanced Force Particle Model", we learned about Newton's first and third law, free body diagrams, balanced free body diagrams, force vectors, weight and mass, friction, as well as different forces acting upon objects in motion, at rest, and accelerating.

Free Body Diagrams: Free body diagrams are a way of illustrating forces that are acting upon an object. It is important to remember that when drawing a diagram, we only include forces acting upon an object and not forces that the object is creating. Several important rules are used when creating a free body diagram.

Newton's 1st Law: Newton's first law states that an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. One real world example of this was when we rode the hovercraft. We continued to move without anyone pushing us. 

(A video of a hovercraft)

Newton's 3rd Law: This law states that for every action, there is an equal and opposite reaction. The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object. For example, when I push on the wall, the wall pushes back with the same amount of force. A real world example is when a car hits a bug on the road. Although it seems counterintuitive, the bug pushes back on the car as hard as the car pushes on the bug. This is a good example of an equal and opposite reaction. 

Both the push on the wall and the push back from the wall are equal.

Friction Force: Friction is the resistance that one surface or object encounters when moving over another. This means that it often slows an object or keeps it from moving in a certain direction. It is always parallel to the surface the object is on.

Gravity Force: Vector always goes straight down, no matter which way the axis is tilted.

Normal Force: Vector always goes perpendicular to the surface the object is resting upon.

Tension: Used when a rope is present, goes in the direction that the rope is pulling.

Tilting an Axis: When an object is on a surface that is at an angle, we can tilt the whole coordinate plane in order to make the free body diagram easier to draw. The x-axis can be tilted to be at the angle of the surface.

Vectors: Vectors are a special type of arrow used in physics. A vector's length determines how large it is in value. For example, if one vector is longer than another, it will have a larger force than the smaller one.

Splitting Vectors: Some vectors will not land on the x or y axis. This is a problem, because we can only balance a diagram when its components lie on an axis. To do this, we can break the vector into x and y components. The vectors on the x and y axis will represent this force in a way that we can use to balance an equation.

Balanced Diagrams: When vectors are equal length on opposite sides of the same axis, the diagram is balanced. This means that the object is either at rest or moving at a constant velocity. When vector lengths are not equal, an object is speeding up or slowing down.


In the example below, a box is resting on a slope. Because the surface that the box is on is tilted, we can tilt the axis in order to create an accurate free body diagram. Using our rules above, we can conclude that gravity pulls the object straight down, normal force goes perpendicular to the surface, and friction keeps the box from sliding down the hill. One thing that you will notice when looking at the diagram is that gravity does not lie on an axis. Because of this, we must split it into two vectors, Fgx and Fgy as seen below. Also, the FN and Fg vectors are much longer than the Ft and Fgx vectors. This tells us that the voce of gravity and normal force are larger than that of friction and gravity on the y-axis. We can also conclude that the object is either at rest or moving at a constant velocity because of the equal vector lengths.

Below is an example of a diagram for an object that is slowing down. This particular diagram is for a ball being thrown in the air. Notice how the only force is gravity, which means that this equation is unbalanced. This means that the object is either accelerating, changing direction, or slowing down. 

Weight and Mass: Although commonly thought of as the same thing, weight and mass are actually very different. An equation to find the weight or mass of an object is Weight = (Mass)(Gravity). Gravity is always 10 (on Earth), weight is always measured in Newtons, and mass is always measured in Kilograms. 

Example problem: A ball has a mass of 15kg on earth. What is its weight? 

-W=(mass)(gravity)

-W=(15)(10)

-W=150 Newtons

Kinetic Friction: Friction can be determined with the equation f=(µk)(W) Where F is frictional force, µk is the coefficient of friction, and W is weight. Friction never changes with speed or surface area, but will change with different types of surfaces as well as with different weights. 

Example problem: The coefficient of kinetic friction is .5 between an 80N book and sandpaper. What is the force of friction between the book and the sandpaper when it is sliding? 

-   f=(µk)(W)

-   f=(.5)(80)

-   f=40N

Solving for Unknown Values In FBD's: In a free body diagram, we often have information like angles and some vectors, but not the vector that we need in order to solve for a certain force value. To solve for a given side, we can use Sin, Cosine, and Tangent. By using the mnemonic "SohCahToa". In the diagram below, angle X has an opposite side, adjacent side, and hypotenuse. These can be used to determine the length of certain vectors. 

   

                                     

              

Example: 

By using our knowledge of sin, cosine, and tangent, we can determine that we must use sin, since we are using both the opposite and hypotenuse.

Work: Sinθ = Opposite/Hypotenuse

Sin (40) = 9/x

 (X)(sin40) = 9

X = 9/(sin40)

x = 12.001

Real World Connection: We encounter forces every day. Although we may not notice it, we are constantly exerting forces in order to move and do everyday chores. For example, as you are reading this, gravity and normal force are acting upon you right now. Action reaction pairs, and Newton's laws are also constantly appearing when we drive, have tug of war contests, and when we are sleeping. In this unit, we used real world situations and explored the reasoning and math behind them.

Texting while driving lab

Unit 1 Summary Blog Post

    

What did I learn throughout this unit?

Throughout this unit, we have learned about position, slope, velocity, average velocity, distance, displacement, linear and parabolic graphs, intersections on graphs, as well as dependent and independent variables. With some explanation, these scientific terms are given a whole new meaning that is very relevant to our lives on a daily basis. Below I will define these terms and explain how they relate to physics and the modern world.


Constant Velocity Particle Model:

X = (V)(t) + Xo

This states that position = (velocity)(time) + starting position. This equation is often used in physics to help us estimate and calculate values on graphs.


Dependent/independent variables:

-x axis = independent variable (the variable we control as scientists) 

-y axis = dependent variable (the variable we measure)

Position: Position is where something is located. We use graphs and axes to give the position of something a number, although the number is only assigned to give a numerical explanation to where something is located. An example of this is where you are sitting right now, reading this blog. Although your "position" does not have a distinct numerical value, we can assign it one in order show where you are on a graph. Position on a graph is defines by the variable "X".


Velocity and Average Velocity: 

Velocity and average velocity have two different meanings. Velocity, how fast something is traveling at a certain point, can be shown through the slope of a graph. It is important to remember that when considering velocity, we must know that velocity can be negative or positive. Slope, measured by the equation (ΔY) / (ΔX) is equal to the velocity of an object. Average velocity, on the other hand, is shown by (Total change in X) / (time). Both Velocity and average velocity are depicted in the graph below. 

                                                               

In this graph, we can use the equation (ΔY) / (ΔX) to determine the velocity between six and eight seconds. in the graph, between the sixth and eighth seconds, the change in Y is -4m and the change in X is 2s. Thus, the velocity is -4m/2s which equals -2m/s. The average velocity on this graph is shown by (Total change in X) / (time). The graph both starts and ends at zero, giving us a total change in position of 0m. Our change in time was eight seconds. By plugging this in, we get 0/-8, which equals zero. The average velocity, although we moved, is still 0 on this graph due to the same starting and ending position. 



Distance and Displacement: 

Distance vs displacement can be depicted on the diagram to the right. Distance, or path route, is the total amount ground covered. For example, on the diagram on the right, you may have walked on a path through a park shown by the blue line. This path was 500m long, known as the distance. Although you have walked 500m on this path, you have only really traveled 250m from your starting point. You may have covered more ground to get there, but the direct route between start and end point was only 250m and is called displacement. Distance can be found by measuring the route an object took while displacement is measured by using the equation Displacement = X final - X initial (X final and initial being end and start points). 


Linear Graphs: 

Linear graphs are depicted by a straight line on a graph. These lines are very useful for having a visual depiction of data The two linear graphs that we use the most in physics are the velocity vs time graph and position vs time graph. Below are examples of both of these graphs and an explanation about each graph and the information they give us.



       

The velocity vs time graph on the left shows us

 how fast an object is moving at a particular 

time. These graphs are created with the equation 

Velocity = (Position) / (Time) For example,

in the graph above, the velocity is on the y-axis and 

the time is on the x-axis. Between zero and two

seconds, the object is moving at a velocity of 2m/s.

Then, for the  next two seconds it has a velocity of 

0m/s, a period of no movement. For the final two 

seconds, it has a velocity of -2 seconds, which means 

that it is traveling back toward the way it came.

         

Similarly, a position vs time graph shows us the 

position of an object after a certain amount of 

time.  These graphs are created with the equation 

Position  = (velocity)(time) + starting position. 

For example, the object starts at 0m but after 5s

it has reached 10m north of the origin. The linear

shape of the graph shows us that the object has

a constant velocity and allows us to predict values. 

Motion Maps:

Motion maps, a way to illustrate the relationship between time and velocity, are very helpful. Each second in a motion map is shown by a black dot and distance is shown by the axis that the motion map is plotted on. For example, in the motion map below, the object starts at 0m and then travels all the way to 6m in three seconds. It then pauses for two seconds and returns to the staring point in two seconds. From this motion map we can use the equation Change in position/time = velocity find the velocity of an object which for the first three seconds is 6m/3s = 2m/s.

Parabolic Graphs: 

Graphs that make a parabolic shape like the one depicted to the right have a variable that increases exponentially. This means that as one variable increases, the other increases at a rate that grows over time, creating a curved line. One example of this was in the rubber band lab. When we pulled the rubber band farther back, the distance it traveled grew exponentially. In order to use these graphs, we must change them into linear graphs. To do this, we must either square a variable or put a variable under 1 (1/X).

For example, in this graph we must square the x-axis variable in order to make it linear. Once we do this, the graph becomes linear and we can make predictions, but, we must remember to use the new variable X^2 in our equation.






Graph intersections: 

In our buggy lab, we learned that when two objects start away from each other and move towards each other at a constant velocity, we can predict intersection point from a graph as well as algebraically. The intersection point of the two lines is equal to the intersection of the two objects.

Equations: (buggy A) X = -30(t) + 200
                  (buggy B) X =  43.84(t) - 230 

Since we want the objects to have the same position (intersection), we set the equations equal to each other and get -30t+200 = 43.84 -230. By solving for t, we can get that the amount of time before an intersection would be 5.75 seconds. Since we are solving for position instead of time, we plug this time back in to the equation to solve for the position of the intersection. X = -30(5.75) + 200 which solves to be 27.6cm north of the origin, also represented by the intersection of the two lines on our graph.









Part B: Real Life Connections

In everyday life, we use velocity, intersection points, and the constant velocity particle model. Think about it, when meeting a friend by car, your meeting point is your intersection point. When predicting when we would meet, we are using a form of the intersections of two lines equation without knowing it. Also, when predicting how long it would get to school we use a form of the constant velocity particle model in order to predict how long it takes to go so far in a certain amount of time at a certain speed. Everyday we are using the connections we made in this unit without realizing it. After studying this unit, I am amazed at how common these ideas are in my daily life.