Website intro to vectors - honorsPresentation Transcript
G WHIZ LAB• Grades are curved: graded out of 28 points instead of 33 Does a more massive object fall faster than a less massive one?Were you able to calculate acceleration due to gravity close to the accepted value?
LAB REPORT COMMENTS Experimental DesignProblem Definition Section • Materials and safety notes• Problem statement: What are we trying • Make sure to list at least 2-3 safety notes for to do in this lab? the lab • Constants in this lab? • Prove the accepted value of gravity: 9.81 m/s2 • Time interval, dot timer used, if person dropping/holding dot timer were the • See if mass affects acceleration same, possibly length of tape was held constant due to gravity • Some people put gravity as constant – this is• Hypothesis should address the what we are trying to prove in this lab though problem statement(s) • Procedure should be stated in sufficient detail so that it could be reproduced • What side of the dot timer were the washers? • How was the dot timer held when paper was dropped through? • Draw pictures of procedures if necessary
LAB REPORT COMMENTS• Data Presentation Section • Include a “data presentation” section in report. Tell reader to see attached tables & graphs if they are attached at the back of the lab report • Tables • Add a border and center text for tables done on excel • Title tables • Calculations & Equations used • Include this information here instead of in discussion & conclusion section • First show equations, then sample calculations using those equations • Show one sample calc for all calculations done • Graphs • Make sure to give a descriptive title to graphs (not just the default title given by excel)
Conclusion • Be explicit about how IV affected DV…”The IV, which was the # of washers, had an affect on the DV, the distance between LAB REPORT COMMENTS dots…” • Refer back to your hypothesis: were your right? Discussion Questions • Refer to specific data collected: a specific table or data point in graph; use this information to support your conclusion• Make sure to answer questions so that reader knows question without looking at question. • Validity: Can someone reproduce this lab and get the same results? Were there major errors that invalidate your data?• Use data to support your answers • Were the # of trials appropriate for this lab?• Do not show calculations in this section, just • Were the procedures followed through consistently ? Were things held constant that should have been? Or refer to data to support your answer. should other things have been held constant that were not?• #4: Must divide the time by 60. What should the area under the velocity vs. time graph give • What errors may have occurred? How can you prevent them? you? • What improvements would you make in the lab in the future to make the lab more valid/repeatable? How would you extend• Many people did not even calculate this lab? acceleration due to gravity – must calculate it • Wrap up lab, and connect this lab to an everyday situation. in order to do #1 and 5. • Some people did not even tell what their calculated gravity values were at all in the report! Must calculate it and discuss your results!
UNIT 3: VECTORS& PROJECTILEMOTION• How would you describe to someone how to get from MHS to Catsup & Mustard?
Scalar MagnitudeSCALAR ExampleA SCALAR is ANY quantity Speed 20 m/s in physics that has MAGNITUDE, but NOT a Distance 10 m direction associated with it. Time 25 secondsMagnitude – A numerical value with units. Heat 1000 calories*a scalar item in your text is written in italics. s, d, m, t
VECTORA VECTOR is ANY quantity Vector Magnitude in physics that has BOTH & Direction MAGNITUDE and Velocity 20 m/s, N DIRECTION. Acceleration 10 m/s/s, E Force 5 N, West* A vector quantity in your textbook are denoted in We will use an an ARROW above the variable bold text to show a variable is a vector. The arrow is used to convey direction and magnitude. v, x, a, F
VECTOR Example 1: Mike skipped towards the east at 25 meters/second.• Drawing pictures of physical situation is very helpful when solving vector problems• Vectors represent by arrows • Point in direction of vector Scale: 1 block = 1 m/s • Length of arrow = magnitude of vector Example 2: Taylor pranced north for 40 meters. • Use a scale to do this – usually use a ruler Scale: 1 block = 1 m
APPLICATIONS OF VECTORSVECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them.• Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started? 54.5 m, E + 30 m, E Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn 84.5 m, E conveys DIRECTION. This is the resultant displacement
APPLICATIONS OF VECTORSVECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT.• Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started? 54.5 m, E - 30 m, W 24.5 m, E
a2 + b2 = c2 ORNON-COLLINEAR VECTORS Opposite2 + adjacent2 = hypotenuse2• Vectors have both horizontal and vertical Sinθ = opposite components hypotenuse • Horizontal is usually the east or west directions, right or left Cosθ = adjacent hypotenuse • Vertically is usually the north or or down tanθ = adjacent• Use pythagorean theorem and opposite trigonometry (SOHCAHTOA!) to solve hypotenuse opposite Θ adjacent
NON-COLLINEAR VECTORS: DRAW A DIAGRAMWhat do vectors look like graphically in physics? A man walks 95 km, East then 55 km, north. The hypotenuse in Physics is Finish called the RESULTANT. 55 km, N Vertical Component Horizontal Component 95 km,E Start NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components TAIL TO TIP. The LEGS of the triangle are called the COMPONENTS
NON-COLLINEAR VECTORS: SOLVING MATHEMATICALLYLet’s solve this problem: A man walks 95 km, East then 55 km, north.Calculate his RESULTANT DISPLACEMENT. When 2 vectors are perpendicular, you must use the Pythagorean theorem. Finish RESULTANT. c 2 = a 2 + b2 ® c = a2 + b2 55 km, N c = Resultant = 952 + 552 Vertical Component c = 12050 = 109.8 km Horizontal Component 95 km,E Start
BUT……WHAT ABOUT THE DIRECTION?In the previous example, DISPLACEMENT was asked for and since it is a VECTOR we should include a DIRECTION on our final answer. N W of N E of N N of E N of W W E N of E S of W S of E W of S E of S S
BUT…..WHAT ABOUT THE VALUE OF THEANGLE??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle. SOHCAHTOA! Sinθ = opposite hypotenuse hypotenuse opposite Cosθ = adjacent hypotenuse tanθ = adjacent opposite adjacent
BUT…..WHAT ABOUT THE VALUE OF THEANGLE??? To find the value of the angle we will use a Trig function called TANGENT. 109.8 km 55 km, N opposite side 55 Tan 0.5789 N of E adjacent side 95 95 km,E Tan 1 (0.5789) 30So the COMPLETE final answer is : 109.8 km, 30 degrees North of East
VECTORS: SOLVING PROBLEMS GRAPHICALLYWe have been solving vector Example: A man walks 95 km, East then 55 km, north. problems mathematically, Calculate his RESULTANT DISPLACEMENT and angle. but they can also be solved graphically, using a ruler and protractor.Steps:1. Develop a scale to use to draw the problem.2. Draw the vector graphically.3. Solve for unknowns, using a protractor and ruler.
HOMEWORK SOLUTIONS: SOLVING GRAPHICALLYPage 1: Using a ruler and protractor to find • Page 2: Finding the resultant vectorshorizontal and vertical components of a vector: given its components: • Vertical: 10.8 cm 1. 6.4 cm at 51 degrees above Horizontal • Horizontal: 10.0 cm 2. 11.7 cm at 59 degrees below the • 44.5 degrees horizontal • Magnitude: 14.2 cm
EXAMPLE: BREAKING A VECTOR INTO ITS COMPONENTS Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components? H.C. = ? The goal: ALWAYS MAKE A RIGHT TRIANGLE!V.C = ? To solve for components, we often use the trig functions 25° 65 m sine and cosine. adjacent side opposite side cosineq = sineq = hypotenuse hypotenuse Rearranging these equations to solve for the horizontal and vertical components… adj = hypcosq opp = hypsin q adj = V.C. = 65cos25 = 58.91m, N opp = H.C. = 65sin 25 = 27.47m, E
EXAMPLE A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bears displacement.Step 1: Draw a diagram of the bear’s displacement. 23 m, EStep 2: Find the resultant displacement in the north - =direction & east direction. Draw This triangle.Step 3: Solve for the bear’s magnitude and direction. 12 m, W - = 6 m, S 14 m, N 20 m, N 35 m, E R 14 m, N R = 14 2 + 232 = 26.93m 14 Tanq = = .6087 23 23 m, E q = Tan -1 (0.6087) = 31.3 The Final Answer: 26.93 m, 31.3 degrees NORTH or EAST
EXAMPLE 1: SOLVE FOR RESULTANT VECTORA boat moves with a velocity of 15 m/s, N in a river which flows with a velocity of 8.0 m/s, west. Calculate the boats resultant velocity with respect to due north. Rv 82 15 2 17 m / s 8.0 m/s, W 15 m/s, N 8 Tan 0.5333 Rv 15 Tan 1 (0.5333 ) 28 .1 The Final Answer : 17 m/s, @ 28.1 degrees West of North
EXAMPLE: RESOLVE A VECTOR INTO ITS COMPONENTS A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the planes horizontal and vertical velocity components. adjacent side opposite side cosine sine H.C. =? hypotenuse hypotenuse 32° adj hyp cos opp hyp sin V.C. = ? 63.5 m/s adj H .C. 63.5 cos32 53.85 m / s, E opp V .C. 63.5 sin 32 33.64 m / s, S
EXAMPLE: ADDING TWO VECTORS AT DIFFERENT ANGLES.A storm system moves 5000 km due east, then shifts course at 40 degrees North of East for 1500 km. Calculate the storms resultant displacement graphically, and mathematically. adjacent side opposite side cosine sine 1500 km hypotenuse hypotenuse V.C. adj hyp cos opp hyp sin 40 5000 km, E H.C. adj H .C. 1500 cos 40 1149.1 km, E opp V .C. 1500 sin 40 964.2 km, N 1500 km + 1149.1 km = 2649.1 km R 2649.12 964.2 2 2819.1 km 964.2 Tan 0.364 2649.1 R 964.2 km Tan 1 (0.364) 20.0 2649.1 km The Final Answer: 2819.1 km @ 20 degrees, East of North
MEASURE YOUR REACTION TIME! Reaction time affects your performance in many things that you do in life…so… Today you will determine your reaction time!1. Have a friend hold a meter stick vertically between the thumb and index finger of your open hand. Meter stick should be held so that the zero mark is between your fingers with 1 mark above it. Do not touch meter stick, let it fall freely. Your catching hand should be resting on a table2. Without warning, your friend will drop the meter stick so that it falls between your thumb and finger. Catch the meter stick as quickly as you can!3. Record the distance the meter stick falls through your grasp. Do this five times.4. Calculate your average reaction time from the free fall acceleration and the distance you measure.