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Transcript
WHAT’S THE SHAPE? LOOK AT THE VECTOR,AND THE AXES. WHAT SHAPE CAN YOU MAKE? Original Vector Y component Vector VY X component Vector VX Component – Means to be a piece, or apart, of something bigger A Component Vector is a smaller vector that is apart of a larger one ANALYTICAL METHOD • Use trigonometry to break vectors into smaller vectors called Component Vectors. – The X component Vector is along the X axis – The Y component Vector is along the Y axis • Trigonometry deals with angles, and triangles. • This makes addition easier. • Please remember Trig. is your friend. • We can add these component vectors to find the resultant. TRIGONOMETRY Opposite • • • • Adjacent SINE = opposite/hypotenuse COS = adjacent/hypotenuse TAN = opposite/adjacent SOH - CAH - TOA BREAKING INTO COMPONENTS Use Sine and Cosine: Vector (V) x component (VX) y component (VY) Sine = Y/V VY = V*Sine Cos = X/V VX = V*Cos 1) Cosine closes the angle 2) Sine skies, Cosine coasts BREAK INTO COMPONENTS First Nowuse thatsine, the original and cosine vector to break are broken the vectors into into their smaller components pieces the original two vectors are gone Adding the components Resultant When we can draw thethese resultant weeasily get a right Now, we add vectors by thetriangle, tip tail method we get a right triangle EVERY TIME WE DO THIS. FINDING THE RESULTANT Means Angle Y Total (YT) R X Total (XT) Use Pythagorean Theorem to find the magnitude VR2 = XT2 + YT2 VR = XT2 + YT2 To find the direction use the tangent Tan R = YT/XT R = Tan-1(YT/XT) Solving a problem 20 meters at 30 degrees + 10 meters at 60 degrees VY = 10m *Sin 60 VY = 10m *.866 VY = 8.66 m 10 m 60O 20 m 30O Vx = 20m *Cos 30 Vx = 20m *.866 Vx = 17.32 m Vx = 10m *Cos 60 Vx = 10m *.5 Vx = 5 m VY = 20m *Sin 30 VY = 20m *.5 VY = 10 m Nowuse that the and original vector are broken into into First sine, cosine to break the vectors smaller pieces the original two are gone their components VY = 8.66 m VY = 8.66 m Resultant 10 m + 8.66 m = 18.66 m VY V = x10 = 5mm V17.32m x = 17.34 m+ Vx = m 5m 5m = 22.34 VY = 10 m Now that weahave these component vectors we lengths can addofthem Once we have the right triangle we can find the Now we have regular right triangle and by using trigonometry Tip Tail and form a right Triangle each side by simply adding the vector length’s together we can find the resultant a = 22.34m b = 18.66 m C= resultant Resultant 18.66 m C2 a2 +b2 22.32 m = C2 = (22.32m)2 + (18.66m)2 C2 = (498.18m2) + (348.2m2) C2 = 846.38m2 C = 846.38m2 = 29.09m = tan-1 (b/a) = tan-1(18.66m/22.32m ) = tan-1 (.836) = 39.9O The final answer is 29.09 meters at 39.9O WOW, That a lot of work!! (Is there an easier way, to do this?) Yes, there is a much faster and easier way, called the “box method” The box method is a technique that organizes all the work into a few simple steps that anyone can do Lets look at the same problem again using the box method, on the side will be the longer method In the end chose the method that works best for you. The box method Step 1: Make a box like this one the top, and bottom rows are always done this way Vectors You will need 1 row for each Vector. This problem has 2 vectors so we need 2 columns Total Vx = Vcos () Vy = Vsin () Step 2: Fill in the vector columns (this sets up the problem) 20 meters at 30 degrees + 10 meters at 60 degrees Then use Write The the Vector’s Cosine Length And sine and Equations Angle To fill in In thethe next Vector two column columns Vectors Vx = Vcos () Vy = Vsin () 20 m at 30O 20m*cos(30O) 17.32 m 20m*sin(30O) 10 m 10 m at 60O 10m*cos(60O) 5m 10m*sin(60O) 8.66 m 22.32 m 18.66 m Total Finally add up the two Columns Using the total columns gets us the right triangle again Vx total = a = 22.34m Vy total = b = 18.66 m C= resultant Resultant 18.66 m C2 a2 +b2 22.32 m = C2 = (22.32m)2 + (18.66m)2 C2 = (498.18m2) + (348.2m2) C2 = 846.38m2 C = 846.38m2 = 29.09m = tan-1 (b/a) = tan-1(18.66m/22.32m ) = tan-1 (.836) = 39.9O The final answer is 29.09 meters at 39.9O