Module Four Summary and begin Popsicle stick catapult lab

Wrapped up module four with test of problems concerning the addition of vectors.

The main take-away from vector addition:
If they ask to add two vectors, you have to get the x and y components of each vector and add together.  Then use the Pythagorean theorem to solve for the magnitude.  Then use the inverse tangent to get the angle.   If they give you one vector and ask for the x and y components you take the sine and cosine of the angle times the magnitude.


We made catapults.  Our plan is to use these math calculations next week to analyze parabolic
motion.

Today we used out catapults to shoot marshmallows.  We measured the height and length of the motion.  We measured the time elapsed from launch to landing.  We measured the angle of the launch with a protractor.  I challenged the students to calculate the exit speed of the marshmallow.  Several had the idea to use x = Vinitial x time plus 1/2 at^2.  There are several other calculations required before this equation can be used.  They will have to follow the Dr. Math equations referred to in the previous paragraph.

Several students had difficulty getting consistent trajectories or had launches that went straight up.  We had to experiment with placement of the fulcrum, attachment points of the lever, and the attachment of the spoon to the stick.  Do not bend the spoon; that interferes with the measurement of the lever and the launch angle.  When we were making the catapults I said to use only rubber bands and not tape.  Those who used tape had difficulty making the adjustments needed to fix the trajectory of their launch.

Two Dimensional Motion

Motion can have two forces.  For example, the current of a river acting upon the direction of a boat or the cross winds that impact the direction of an airplane.  To understand the net impact of two forces on an object we have to use basic trigonometry.  We reviewed the basic types of triangles: Scalene, Isoceleses, Equalateral, Obtuse and Right.  We reviewed that the sum of the angles in a triangle is 180 degrees.  We reviewed the difference between Radians and Degrees.  There are two times Pi radians in a circle.  This is important to understand because we will be using TI83 & 84 calculators and also Excel to calculate Sine, Cosine and Tangent.  One has to check the Mode in the calculators to ensure you are using Degrees and not Radians.  In Excel you have to convert degrees to radians before performing the calculations.  I introduced the students to SAHCOHTOA, an anagram for the basic trigonometric equations.
     SAH sine = adjacent over hypotenuse
     COH cosine = opposite over hypotenuse
     TOA tangent = opposite over adjacent

For practice, we measured the height of the library roof.  Each student stood back from the library wall at least 6 to 10 meters.  Then we used a protractor to measure the angle from where the student was situated to the top of the roof.  We therefore had the angle and the adjacent side of a right triangle.  To determine the height of the roof, which is the height of the library wall and is the opposite side of the triangle, we used the tangent calculation.
   

Apologia Physics: Introduction to Gravity or Ball Drop Lab

Ball Drop Lab

Apologia Physics: Motion in one direction

Using formulas to solve for Velocity, Distance, Acceleration, and Time
It is important to remember one key rule of algebra: order of operations, inside parenthesis and outside.  For example in the equation x = 1/2 at^2  be sure to square the number and the units of measure that are represented by t.  It is also helpful to write fractions (and the accompanying units of measure) with the numerator on top of a line and the denominator below the line.  This is obvious but if you use a slash as in the way I wrote one-half above, it can be confusing to eliminate the units of measure.

Several students were showing a shortfall in their knowledge of algebra and scientific notation so I sent them links to Khan Academy's Algebra Basics . Dr. Khan does a great job explaining anything mathematical.  I've used this site to refresh my memory on how to use sine, cosine, and tangent which we will be using soon in our calculations.

We have several students who would rather not use scientific notation but write out the number.  This is okay for small calculations and numbers but will be a major problem when using larger numbers and more complicated equations.  I told the class that if anyone has an aversion to the use of scientific notation they should immediately expunge this thought from their brain!

Measuring displacement using time and acceleration of gravity

This week we have been reviewing the five basic formulas for One-Dimensional Motion and Free Fall.

First lab was to measure the time it took for several objects to fall from a measured height.  We discussed what factors we could control such as using the same person to do the drop and the same person to do the timing.  We used the formula x=1/2*a*t2 to determine if we had measured the time accurately as we knew the displacement (x) and the acceleration of gravity (a).  We calculated the margin of error in our time measurements.

We also had a lesson in solving (using Solver in TI 84) this type of equation on the Texas Instruments graphing calculator.  We have several used graphing calculators.  A good graphing calculator app for ipad (Search for TI84) is about $10, much less than a graphing calculator.  A used TI 83 or 84 is maybe $25. Don't buy an 86 or 89. It would be very useful if you purchased either a calculator or the app as we will be performing many of the same calculations and also need a way to look up sine, cosine and tangent later in the course.

Second lab was using a tennis ball and stopwatch to measure the height of the library roof where we meet.  Again, using the formula above we measured the time for the ball to drop from the roof to the sidewalk.  We were not able to climb on the roof so we thru the ball in the air so that it reached the level of the roof then timed the drop to the sidewalk.  With our measurement of time (t) and the acceleration of gravity (a) we measured the height of the roof.  We were able to verify this by counting the number of panels in the wall of the building and multiplying by the height of a panel.