CAPM Final Challenge - Fan Cars

Fan Cars

What questions did you ask?

1. What formulas will I use?

 - acceleration = change in velocity / change in time

 - velocity = change in position / change in time

2. What do we need?

 - timer

 - measuring tape or a yard stick

 - markers

 - the fan car

3. What measurements do we need to take?

 - position over time

 - velocity over time

*we also need the other group's data so we can make our prediction.

Summary

We began by setting up how we were going to record our data.  We lined up a meter stick on the floor, turned on the fan car, and recorded its position for every second.  We did three trials to make sure that our data was accurate, and we took an average.  We then took the averages from Will and Robert's car, and compared it to ours.  We created a graph using excel to find the point of intersection on the position versus time graph, and we concluded that the cars would potentially intersect after 2.48 seconds at the 60.01 centimeter mark.  Below is the graph and table we constructed to find our prediction:

We created our prediction with our car beginning at the 0 mark on the meter stick, and their cart beginning at the 100 mark on the meter stick.  This allowed for us to view the intersection, as one was going in the positive direction, and one was going in the negative direction.

Our prediction

Time: 2.48 seconds

Position: 60.5 cm

Actual Test

Time: 2.9 seconds

Position: 63 cm

Percent Error

percent error = actual-predicted / actual

Our percent error was 4.7%!!!

Unit 3 - Constant Acceleration Particle Model

Constant Acceleration Particle Model

When an object is moving at a constant speed, it usually looks like this:

This is showing no speeding up, slowing down, or change in direction while the object is moving.  For example, if you were to push a bowling ball with a broom, and let it roll, it would continue to roll at a constant speed in the forward direction.

When an object is accelerating, there is an increase in the speed of the object.  For example:

Acceleration:

A change in the velocity over the change in time.  The acceleration is measured in meters per second squared (m/s^2).  The unbalanced force causes an increase or decrease in speed, or a change in direction.  The acceleration of the object is essentially how fast it is getting faster.  On a graph, a steeper slope shows a greater acceleration.  

Instantaneous Velocity: 

The average velocity for the entirety of a position v.s. time graph  is equal to the instantaneous velocity of the object at the midpoint of the graph.  In order to find the instantaneous velocity for an object at 5 seconds, you would have to calculate the average velocity from seconds 4 and 6.  ( xfinal - xinitial / tfinal - tinitial )

FORMULAS

Slope of a line ....... x final = velocity * change in t + xinitial 

Displacement on an x v.s. t graph ....... change in x = x final - x initial

Velocity on an x v.s. t graph ........ v = change in x / change in t

Acceleration ....... a = change in v / change in t

Displacement ....... change in x = 1/2 acceleration * (change in t)^2 + velocity initial * time

X V.S. T GRAPH QUESTIONS

a) What do they show you about the movement of the object?
- It shows you what direction the object is moving in, and whether it is speeding up or slowing down.

b) What does the shape tell you about the velocity of the object?
The shape shows if it is speeding up or slowing down.  If it is a steep slope, then it has a high velocity, and if it isn't a steep slope, the object is moving with a slow velocity.

c) What's the y-intercept of a x v.s. t graph?
x0 (x knot)

d) How do you use this graph to find the displacement?
Take x final  - x initial to find the displacement

V V.S. T GRAPH QUESTIONS

a) What do these graphs show you about the movement of an object?
- If the object's velocity is going in the opposite direction of the origin, then it is speeding up.  Likewise, the closer the line gets to the origin, the slower the object is moving.
- Constant velocity is shown with a straight line.

b) How do you determine the object's position with this graph? 

- The object is moving forward if the velocity's line is moving north.
- The object is moving backwards in the velocity's line is moving south.

c) How do you calculate the displacement?
- To calculate the displacement, you take the area of the space that is between the line and the origin.

A V.S. T GRAPH QUESTIONS

a) What do these graphs show about the movement of an object?
It shows the rate of acceleration of the object, if acceleration is present.

b) How do you determine the position of the object on an acceleration time graph?
- On a v v.s. t graph, if the slope is positive, the object has a positive acceleration.  This means the acceleration line will be above the origin.  Likewise, if the object has a negative slope on the v v.s. t graph, then the object has a negative acceleration.  This means the acceleration line will be below the origin.  

EXAMPLES

a.

The object is moving in the positive direction, however it's position versus time graph is curved.  This shows the acceleration of the object.  It's velocity over time is increasing as time passes, therefore it is speeding up.  Since it is moving in the positive direction, its velocity is positive, as well as its acceleration.  It's acceleration is constant because it is continuously speeding up.

b.

The object is constantly slowing down, meaning it's velocity's line is moving towards the origin in the positive direction.  It's acceleration is positive and constant, because the object is above the origin.

c.

The object initially has a quick speed, however it slows down and eventually stops.  This is represented in the v v.s. t graph because the velocity's line is going towards the origin.  There is still acceleration, however it is just negative.  If the object's velocity is at a negative slope, the acceleration is underneath the origin.  

d.

The object is moving in the negative direction the whole time, but it is still accelerating.  The velocity is increasing over time, which is shown by it moving further away from the origin.  Since the velocity is at a negative slope, the acceleration is underneath the origin.

PRACTICE PROBLEMS

a) Construct a motion map for the object's velocity and acceleration:.

b) Describe the motion of this object.

The object is constantly accelerating in the positive direction.  It's speed continues to increase as it moves.  

c) Determine the instantaneous velocity for the object at t=2s.  Explain.

v = change in x / change in t

v = 18 - 4 / 3 - 1

v = 14/2

v = 7 m/s 

To find the instantaneous velocity of an object at a certain time, take the plot from one second above and one second below the time you want to solve for, and use the change in x over the change in t, meaning the final minus the initial.

d) Assume the initial velocity was 10 m/s.  Determine the acceleration of the object.

a = change in v / change in t

a = 10 - 7 / 3 - 1

a = 3/2

a = 1.5 m/s^2

e) Show a corresponding velocity time graph for the object.

CAPM Challenge 1

CAPM Challenge 1

 Graphs and Data Charts

This graph represents the data that was initially taken for the ball's position over time.  As we observed, the ball is accelerating constantly, as the graph's shape goes into an upward slant.

In order to better interpret the data, we had to square the time to get a straight line.  This gave us a more accurate and readable sense of how much the ball was accelerating.  From our data, the graph's equation is x = 0.0951t^2 + 0.0349.  

Reminders about formulas

y = mx + b (for a straight line)

to straighten a line, you must use y = mx^2 + b

velocity = change in position / change in time

acceleration = change in velocity / change in time

Mathematical Expression

The acceleration of an object is shown by the change in velocity over the change in time.  We used the formula a = v final - v initial / t final - t initial to solve for the acceleration.

Predicted acceleration 

0.1902 m/s^2

How did we get the velocity of the ball at 4 seconds?

By using our formula , we were able to find the velocity of the ball at four seconds.  We first found the position of the ball when t=5, which was found by substituting 5 for t in the equation above.  We then found the position of the ball when t=3, which was also found by using the equation.  The position of the ball at 5 seconds was 2.41 meters, and the position of the ball at 3 seconds was 1.55 meters.  We subtracted those, and divided them by 2 (5-3) to find the average velocity at four seconds, which was 0.427 m/s.   

Were we within 10%?

We were within 10%!  We found the percentage error by doing the (approximate-predicted) / approximate.  We were only 1% off!  The real acceleration was 0.187 m/s^2, and we got 0.1902 m/s^2.

Conclusion

This was an overall interesting challenge, because we were forced to use all of the knowledge that we had accumulated from the last class to solve the problems.  With help from my two partners, Will R. and Isabella, we were able to find some shortcuts and use the new formulas to find the acceleration of objects, and the velocity of objects.  I thought it was really cool to see us solve this physically by using our knowledge, and then compare it to the actual information by using the cart roller.  It's pretty cool to be so accurate without using all of the technology!

BFPM Practicum

BFPM Practicum

The Set Up:

Free Body Diagram:

Workings and Challenge Solving:

Cos θ =  adjacent / hypotenuse                                                               Cos θ  = adjacent / hypotenuse

Cos 53 = Ftby / 0.9N                                                                              Cos 13 = Ftay / 2.2N

0.9 cos 53 = Ftby                                                                                    2.2 cos 13 = Ftay

0.5N = Ftby                                                                                            2N = Ftay

PREDICTED TOTAL : 2.5N

Unit 2 - Balanced Force Particle Model

Bowling Balls and Hovercrafts

To speed up an object...

- Apply more force on it from behind

- The boy continuously pushed the ball with a greater force 

To stop an object...

- Apply force in the opposite direction of travel

- The boy stopped the ball by putting his hand on it

To keep an object moving at constant velocity...

- Apply no force once it is moving

- Nothing needs to keep this object moving as long as nothing stops it

Constant Velocity Misconception

Situation: A cart is moving at a constant velocity of 10mph, and a ball will pop out of it.  Will the ball land in front of the cart, behind the cart, or in the cart?

Explanation: Many people don't understand that the ball is moving with the same velocity as the car.  Once the car starts moving, the car and the ball move forward at the same speed because the ball is in the car.  When the ball is thrown upwards, with the cart continuously moving, the ball moves forward with the cart and lands back inside.  There was no outside sideways force to prevent the ball from continuing on its path of 10mph with the cart.  

Introducing Full Body Diagrams

When systems are balanced, they have an equal amount of force on each side.  There is also no change in the up and down motion.

When systems are unbalanced, they have an unequal amount of force on each side.

Newton's 1st Law 

 When all forces are balanced, an object in motion/at rest stays in motion/at rest unless an outside force acts upon it.

Types of Forces

Normal Force - The force perpendicular to the two surfaces that touch each other

Frictional - The force parallel to the surface at which an object travels

Tension - The force evident when there's a rope/chain exerting tension on an object

Gravitational - The force exerted through a force field by two objects (when the situation is on earth)

FBD Examples

 The ball is not moving, therefore it is resting at a constant velocity.  There is no force going horizontally that is making it stay that way, therefore the only forces on this object are the gravitational force and the normal force, which are going vertically.  (Make sure you use vectors to indicate balance)

The man is pushing the table with a force of 20 N.  By removing friction from this situation, the speed at which he's pushing the table is speeding up.  The forces acting on the table are the push, gravitational, and normal.  

The ball has been thrown into the air.  While it is in the air, the only force acting on it the entire time is the gravitational force.  Whether it be travelling upwards, at it's peak, or travelling downwards, the only force evident is the gravitational pull from the earth.  

The parachutist is moving at a constant velocity after pulling his parachute out.  If the skydiver didn't have the parachute, there would be a greater gravitational force and a smaller air force until it would reach terminal velocity.

Imaging you're in a car that's moving 50mph.  When braking, there is an unbalanced net force pushing the car backwards, slowing the car down.  The seatbelt prevents you from moving forward by providing an unbalanced backwards force.  

Introducing Newton's 3rd Law

For every action, there is an equal and opposite reaction.

Example 1 - A Mercedes Benz going 50mph and a Mac cement truck going 50mph collide on I-40.  Which one was hit with more force?  Why?
Answer - Neither was hit with more force!  According to Newton's 3rd Law, each action has an equal and opposite reaction.  So, if the Mercedes Benz hit the Mac truck with 1000N of force, the Mac cement truck hit the Mercedes with the same amount of force.  
Example 2 - A man is applying a push force on a table in the science lab, but it's not moving!  What is going on?
Answer - Two things are happening in this situation.  Not only is friction working against his push force on the table, but the same amount of force exerted onto the table is being pushed back onto the man.

Tug of War (expanding on Newton's 3rd Law)

When playing a game of tug of war, you may think that the way to win is through having a strong team.  However, whichever team has the most friction tends to win.  

*Keep in mind that both surface area and velocity DO NOT affect friction.

Full Body Diagrams with Slope

The skier on the left is moving in a downward slope on a mountain at a constant velocity.

The free body diagram shows the tilted axis, that's friction force is now parallel to the surface.  We had to split the Fg to make it a balanced diagram.

Full Body Diagrams with Angle

The box is being pulled at a constant velocity.  If the box had a weight of 200N, you could use a formula like cosine = adjacent/hypotenuse to find the Fgy.  The Ft in this situation is the 200N.  

cos x = adj/hyp

cos30 = Fgy/200

200 cos30 = Fgy

31N = Fgy

Finding the Mass of an Object

w = mg

Remember: 

w = weightm = massg = gravity (a constant of 10N)

Example 1 - An object has a mass of 20 kg.  What's the weight of the object?

w = mg
w = (20kg)(10N)
w = 200N

*Reminder: When calculating the weight of an object, the mass has to be in kg, so convert it from grams to kg if necessary.

Finding the Kinetic Friction of an Object

Ff = µk * weight

Texting While Driving

Texting While Driving

Situation

You're driving at 50 mph on I-40 in the middle lane (that's approximately 88,000 yards per hour).

Key Points

- You're travelling at 1,445 yards per minute (24.25 yards per second).

- You're using the iPhone 5s.

- The "texting process" was from opening the lock screen, to opening the messages app and texting LOL.  This was what we timed and used as our data.

- We took 6 trials of data, and it took an average of 7.22 seconds to go through the texting process.

Data

Conclusion

If you were driving at 50 mph, and took your eyes off of the road to text someone LOL, you would have travelled a total of 175.1 yards in the 7.22 seconds it took to text your friend.  We calculated this by finding the average distance that the car would travel per second if it went 50 mph (24.27 yards per second), then multiplied it by the amount of time it took to go through the  "texting process" (7.22 seconds).  During our tests, we made sure to use the same phone, and we made sure the car's speed remained constant.  

Unit 1 - Graph Trends and The Constant Velocity Particle Model

        Graph Trends

Motion Maps

- Represent the position, velocity and acceleration of an object.

In the graph above, it's shown that object B is travelling at a faster rate than object A.  You know this because object B travelled a further distance than A in the same given amount of time.  When the arrows are longer, it implies that the object is moving faster.  

Displacement

The displacement of an object is how far away the object ends up from its starting point.  You are always measuring the distance between two points.  The way to calculate this would be to observe the change (Δ) in position, which would be (x final - x initial).  

Path Length

The path length of an object is the distance it has physically travelled.  So, if it went 5 meters at 1 m/s in the northern direction, then travelled south for 7 meters at 1 m/s, the path length would be 12 m.

Speed

The speed is how fast the object is going on the path

Velocity

The velocity of an object is the change in position over the change in time.  

Practice Problems!

Describe the motion of the object.

The object begins at the 20 meter mark on the y axis, maintaining a velocity of 0 m/s for the first second.  It then travels 20 m/s for two seconds in a positive slope, until it plateaus again at 60 meters.  It obtains a velocity of 0 m/s for a second and a half, before continuing on at a negative slope of about 26 m/s.  

How much distance did the object travel?

It travelled a total of 80 meters.  It began at the 20 meter position, proceeding to the 60 meter position, and eventually going back to the 20 meter position.  

What was the total displacement?

The total displacement was 0, because it started and finished at the same point on the position axis.

Mathematical Equation describing motion

x = vt +x0 <-- (starting position)

Velocity Graphs

A velocity graph is represented with a straight line, no slopes are involved.  The x axis usually relates to the time, and the y axis is the object's distance over time (velocity).  

Determining area

Calculate the area up to 5 seconds in the graph above. 

5 s * 10 m/s = 50 ms/s = 50 meters

This area represents the object's displacement, so the object ended up 50 meters from its starting point.

How does this relate to our world?

Objects in motion always have some form of velocity.  Whether it be the car driving past your house, or the ball being thrown at baseball practice, all objects that are moving have velocity.  Constant velocity is measured by the displacement over time.  If you drove 60 mph for an hour, your constant velocity would be 60 miles per hour.  Ideas like displacement also are seen in our everyday life.  If you began a walk at your house, and travelled around the neighborhood to finish your walk at your house, your displacement would be 0.  There was no change in the starting or finishing position in that situation.  An object in motion also always has some type of speed.  Whether it be going fast or slow, speed can be measured as distance over the amount of time.