Adding Vectors by the Component Method Feel free

Adding Vectors by the Component Method Feel free to use to accompanying notes sheet.

Adding Vectors by the Component Method Yesterday we added vectors which were at right angles to one another. What would happen if the vectors were not at right angles? ¡ A crow (who apparently isn’t aware that he should only fly in straight lines) first flies 10 km at 60° N of E. The he flies 25 km at 10° N of E. What is his total displacement? ¡

Adding Vectors by the Component Method – The Strategy B = 25 km @ 10° N of E 10° A = 10 km @ 60° N of E 60° ¡ Obviously the answer is not 35 km, so what must be done?

Adding Vectors by the Component Method B = 25 km @ 10° N of E R = resultant A = 10 km @ 60° N of E Obviously the answer is not 35 km, so what must be done? ¡ Find the RESULTANT ¡

Adding Vectors by the Component Method The vectors need to be added vectorally. ¡ Both vectors need to be RESOLVED into their components. ¡ The components are then added together to find the resultant. ¡ A to be resolved into Ax and Ay. ¡ B to be resolved into Bx and By. ¡

Adding Vectors by the Component Method – Resolve each vector into components B = 25 km @ 10° N of E A = 10 km @ 60° N of E

Adding Vectors by the Component Method – Resolve each vector into components B = 25 km @ 10° N of E Bx Ay A = 10 km @ 60° N of E Ax By

Adding Vectors by the Component Method B = 25 km @ 10° N of E By Bx Ay A = 10 km @ 60° N of E Ax Ax = 10 cos 60° Bx = 25 cos 10° Ay = 10 sin 60° By = 25 sin 10°

Adding Vectors by the Component Method Bx By Ay Ax ¡ The x-components and y-components can each be considered legs of the resulting triangle.

Adding Vectors by the Component Method By Ay Ax ¡ Bx The x-components and y-components can each be considered legs of the resulting triangle.

Adding Vectors by the Component Method – Construct the “resulting triangle” from components By Ry R Ax ¡ Rx Rx = A x + B x Ay Bx Ry = A y + B y

Adding Vectors by the Component Method Rx = 29. 60 Ry = 13. 00 By R = ? ? Ry R Ax Rx Ay Bx How do we find R? YES!! Pythagorean Theorem: R 2 = Rx 2 + Ry 2

Adding Vectors by the Component Method By Ry R Ax ¡ Rx R 2 = R X 2 + R Y 2 Ay Bx

Adding Vectors by the Component Method Math – IN DEGREE MODE ¡ Ax = 10 cos 60° = 5. 00 km ¡ Ay = 10 sin 60° = 8. 66 km ¡ Bx = 25 cos 10° = 24. 6 km ¡ By = 25 sin 10° = 4. 34 km ¡ Rx = Ax + Bx = 29. 6 km ¡ Ry = Ay + By = 13. 00 km ¡ R = 32. 33 km ¡

Adding Vectors by the Component Method ¡ Another way to organize data x-axis y-axis A 10 cos 60° 10 sin 60° B 25 cos 10° 25 sin 10° R

Adding Vectors by the Component Method ¡ Another way to organize data x-axis y-axis A 5. 00 8. 66 B 24. 62 4. 34 R

Adding Vectors by the Component Method ¡ Another way to organize data x-axis y-axis A 5. 00 8. 66 B 24. 62 4. 34 R 29. 62 13. 00

Adding Vectors by the Component Method Rx = 29. 60 Ry = 13. 00 By R = 32. 33 Ry R Ax Rx Ay Bx Is this it? 32. 33 km?

Adding Vectors by the Component Method Rx = 29. 60 Ry = 13. 00 By R = 32. 33 Ry R Ax Rx Ay Bx Is this it? No!! Now we have to find the direction.

Adding Vectors by the Component Method Rx = 29. 60 Ry = 13. 00 R = 32. 33 R θ Rx tan θ = Ry/Rx θ = tan-1(Ry/Rx) Ry

Adding Vectors by the Component Method Rx = 29. 60 Ry = 13. 00 R = 32. 33 R θ Ry Rx θ = 23. 71° N of E How do we know it is North of East?

Adding Vectors by the Component Method N (north) A W (west) B 40° N of E C Which angle is 10° North of West? A, B or C? Which angle is 10° West of North? A, B or C? What is angle C? S (South) E (east)

Adding Vectors by the Component Method B = 25 km @ 10° N of E R A = 10 km @ 60° N of E ¡ Final answer: 32. 33 km @23. 71° N of E

Adding Vectors by the Component Method ¡ Some helpful hints! l l l Never use the original values after the vector has been resolved. Assign negative values to S and W components of vectors (assuming N and E are positive) Always make sure you are in degree mode.

Adding Vectors by the Component Method ¡ Some helpful hints! l l Always make sure you are in degree mode. Make sure you draw your vectors in the correct directions initially. To help, redraw a coordinate system at the end of each vector. Be organized, stay organized, & finish the entire problem.

Adding Vectors by the Component Method ¡ Other examples l A duck flies 10 m/s @ 30° S of W with a wind blowing 5 m/s N. What is the resulting velocity of the duck? ¡ Answer:

Adding Vectors by the Component Method ¡ Other examples l A duck flies 10 m/s @ 30° S of W with a wind blowing 5 m/s N. What is the resulting velocity of the duck? ¡ Answer: B 0 x & +y A -x & -y

Adding Vectors by the Component Method ¡ Other examples l A duck flies 10 m/s @ 30° S of W with a R = ? ? wind blowing 5 m/s N. ¡ B Answer: x-axis y-axis A -10 cos 30 -10 sin 30 B 0 +5 R A

Adding Vectors by the Component Method ¡ Other examples l A duck flies 10 m/s @ 30° S of W with a R = ? ? wind blowing 5 m/s N. ¡ B Answer: x-axis y-axis A -8. 66 -5 B 0 +5 R A

Adding Vectors by the Component Method ¡ Other examples l A duck flies 10 m/s @ 30° S of W with a R = ? ? wind blowing 5 m/s N. ¡ B Answer: x-axis y-axis A -8. 66 -5 B 0 +5 R -8. 66 0 A

Adding Vectors by the Component Method ¡ Other examples l A duck flies 10 m/s @ 30° S of W with a wind blowing 5 m/s N. ¡ Answer: R = -8. 66 m/s or 8. 66 m/s W B A x-axis y-axis R -8. 66 0

Adding Vectors by the Component Method ¡ ANY and ALL vectors can (and will) be analyzed this way. l l l Displacements Velocities Accelerations Forces Momentum

Adding Vectors by the Component Method Any questions? ¡ These notes will be online ¡ You MUST be good at vectors to succeed / pass this class. ¡ Ask questions (in class) whenever necessary. ¡