Graph vs Graph vs Map

Position vs. Time, Velocity vs. Time, and Motion Maps

1) Position vs. Time

• y-int = starting point
• x-axis = reference point
• flat lines = the object is stopped at one position
• positive slope = object traveling away from reference point
• negative slope = object traveling toward reference point
• slope = velocity (displacement/time)

2) Velocity vs. Time

• flat horizontal lines = constant velocity
• x-axis = object is stopped
• above the x-axis = positive velocity (moving away from reference point)
• under the x-axis = negative velocity (moving toward reference point)
• vertical lines = change in velocity

3) Motion Map

• dot = second 
• arrow = distance traveled in a second
• short arrows = low velocity
• long arrows = high velocity
• dot (no arrow) = no movement
• right arrows = positive movement away from reference point
• left arrows = negative movement toward reference point

Effect of Direction

In class, we had a VERY long, in depth conversation about the effect that direction has on measurements. 

1) Distance & Position

• distance - measurement of total length traveled
• position - displacement from reference point in a certain direction
• displacement - length between starting point and ending point

Unlike distance, position is effected by direction. Distance is also the total length traveled that includes if one travels backward.

Ex) a cyclist rides his bike for four miles, rides back one mile, and then turns back around and continues in the original direction for three miles.

Distance = 4 + 1 + 3 = 8

Position = 4 - 1 +3 = 6

*Distance and displacement are also measured starting at the starting point; while, position always refers to the reference point.

2) Speed & Velocity

• speed - distance/time
• velocity - position(distance+direction)/time

*Speed is the slope of a distance vs. time graph; while velocity is the slope of a position vs. time graph.

Also, the formula for position is...

x    =    v    •    t  ±  x₀

^position ^velocity ^time ^starting position

Buggy Lab

I thought this was kind of a weird lab because we got some odd results for our second scenario.

In the first scenario, we started the buggy car at the reference point and counted how many centimeters for every 5 seconds that the buggy car ran.

This first scenario was pretty easy. Our results gave us a direct, linear graph.

Although in the second scenario, we moved our buggy car back 40 cm behind the reference point. We still counted how many centimeters away from the reference point for every 5 seconds that the buggy car ran.

This is where the results get funky. Our graph is like an absolute value. My group decided on this graph because a position cannot be a negative distance away from the reference point. So for those who don’t know, the graph would look kind of like a linear line and then flipping the negative half of the line over the x-axis to make a “V” shape.

In our graphs, the x-axis was the reference point. Whether at the origin or somewhere along the x-axis, that point where the line touches the x-axis always represents the reference point.  

Converting: English Style

Last time I talked about converting between metric units. This was pretty easy for me (even using powers of 10).

Although this time, I’ll be talking about converting between English units. My friend showed me this little trick on remembering the conversion rate between gallons and cups. It’s called the Gallon Man.

Ain’t it cute!

Anyway, the reason that conversions between English units are harder is because the conversion factors are not standardized and are very random. So, converting between English units takes a longer process than just multiplying and dividing by 10. The easiest way to convert is cross cancellation of units.

Ex)  1m│100cm = 100cm

        1   │ 1m            1

m → cm

The two m (meter) are cancelled out because m/m=1.

Let’s try one a little more complicated

Ex) 200cm│   1m     │  1km  │ 60sec │60min =  720,000km = 72km =   72km/hr

         1sec │ 100cm │ 100m│ 1 min   │1 hr          10,000 hr       1hr

This conversion shows cm/sec → km/hr

Cross cancellation works by multiplying all the top numbers together then dividing it by the product of all the bottom numbers. 

Converting: Metric Style

Most of you probably know how to convert between metric units, but if not, this is how you do it with powers of ten.

Metric units all go up by powers of ten from the base unit (such as liters, meters, and grams). So 1m is equal 10²cm (100 cm), or 1m is equal 10⁻³ km (.001 km). Yes, the power can be positive or negative. It will be positive if the unit you are converting to a unit smaller than the original unit used (m→cm). While, the power will be negative if the unit being converted to is bigger than the original (m→km). The powers I keep mentioning refer to scientific notation. In math, you may have seen something such as 4.5•10². This really means 450. Since the power is 2, you move the decimal 2 places to the right. If the power was -2 (4.5 •10⁻²), you would move the decimal 2 places to the left (.045).

 

So positive = decimal moves right, and negative = decimal moves left

Tile Lab

 

This past class we white boarded yet another lab, the tile lab, from our “laborama” (see first post). In this lab we had to compare the area and the mass of VERY odd, annoying, and random cut pieces of carpet.

1)      Question

·        What is the relationship between area and mass?

2)      Prediction

·        Our group predicted a direct, linear graph with a small gradient. Also, area would be the independent variable(x) and mass would be the dependent variable(y).

3)      Set-Up

·        The set-up of this lab was pretty simple. Since we’re not allowed to “calculate” measurements (meaning we can’t use math to find our measurements), we had to trace the tiles on a piece of graph paper and count the number of squares to find the mass. Then to find mass, we used the triple beam balance.

4)      Data

·        Ok. So obviously if you looked at my group’s data table, you can see that a few of the measurements are really weird and completely off. I feel that some of the problems were due to the fact that we had multiple people trying to count the squares each using a different method.

5)      Graph (& Formula)

·        Our graph is probably really confusing because there are two formulas and one of them is quadratic even though the line is evidently linear. So our group was split between linear and quadratic. The half that thought it was quadratic only picked so because the points lined up well with that equation; while, my half went with linear because we knew that linear was the obvious chose and the linear equation lined up very well without the outliers.

6)      Conclusion

·        On the picture, the conclusion says, “For every cm² that area increases, mass increases by .006 grams”. That conclusion statement was for the other half of the group’s equation. So for our conclusion we got – For every cm² that area increases, mass increases by about 1.8 grams.

Indentifying Direct, Indirect, Proportional, and Inverse Relationships

Last class we had a very in depth conversation about the differences between these relationships. For each of them, we discovered a few unique traits that each must have.

1)      Direct

·        When x increases, so does y

Ø  This means the graph must have a positive slope. Whatever direction x moves, y must move in the same direction even if that means that they both move negatively.

·        Y-intercept

Ø  The graph can start anywhere on the y-axis.

·        Type of graph

Ø  Any type of education can represent a direct relationship as long as it follows the other traits.

2)      Indirect

·        “Not Direct”

Ø  Indirect literally breaks down into “not direct”. So whenever a graph does not follow all of the traits for direct it is indirect.

3)      Proportional

·    Δ x = Δ y   

Ø  In a proportional function, the change in x is equal to the change in y. So if x doubles, then y doubles also.

·        Y-intercept

Ø  In a proportional graph, the y-intercept must start at the origin. A y-intercept would not allow for the ratio between the change in x and y to remain proportional.

4)      Inverse

·        k/x (when k is a constant)

Ø  In this inverse parent function, it shows the opposite of proportional. If x doubles, y is cut in half.

*Also, some functions can be more than one of these relationships at the same time.