1. MOTION IN TWO DIMENSIONS VECTOR ALGEBRA PROJECTILE MOTION RELATIVE MOTION P P P

2. POLAR FORM OF VECTORS: The polar form of a vector simply lists or states the magnitude of the vector together with its direction with respect to a reference. EXAMPLE: 50m/s at 20 o In print the usual symbolic notation for a vector is a capital letter in bold print. In writing the usual notation for a vector is a capital letter with a small horizontal arrow above it.  AorA

3. GRAPHICAL REPRESENTATION OF VECTORS: A vector quantity may be depicted using an arrow, a straight line with a point. The length of the arrow will be proportional to and represent the vector’s magnitude (which is always positive). The orientation of the arrow in space or on a coordinate grid corresponds to the vector’s direction. 50 o This arrow represents the velocity vector of 4 m/s at 50 o. x y

4. GRAPHICAL METHOD OF ADDING VECTORS: [ HEAD - TO - TAIL METHOD ] Add: 20 at 60 o,30 at 135 o, and40 at 340 o x y 

5. COMPONENTS OF A VECTOR: x  = 130 o 50 A x A y  as shown is in standard form.

6. ADDING VECTORS USING THEIR COMPONENTS:

7. Add: A = 20 at 40 o and B = 30 at 210 o C = A + B VectorXY 20 @ 40 o 20cos 40 o 20sin 40 o 15.3212.86 30 @ 210 o 30cos 210 o 30sin 210 o -25.98-15.00 C x = -10.66 C y = -2.14

8. When using a calculator to calculate tan -1, add the appropriate correction angle to the result given by the calculator. USING tan : 2 = tan (C y /C x ) + Correction I C x + : C y + Correction = 0 o II C x - : C y + Correction = 180 o III C x - : C y - Correction = 180 o IV C x + : C y - Correction = 360 o x y

9. Answers on next slide.

11. Answer on next slide.

12. Answer: 60 at 25 o

13. Answer on next slide.

14. x-component: 40 cos 295 o = 16.9 y-component: 40 sin 295 o = -36.3 x y ANSWER:

15. Answer on next slide.

18. Example: In a physics demonstration a golf ball is thrown 70m. Its angle of launch is 45 o. What is its initial speed?

19. ANSWER: In a physics demonstration a golf ball is thrown 70m. Its angle of launch is 45 o. What is its initial speed?

20. Relative Motion: Consider yourself to be a stationary observer. A friend of yours, Joe, is another observer but he is in motion. Lets say you measure his velocity to be the vector u. Some object of mutual interest passes by. Both you and Joe measure its velocity. Joe measures its velocity to be v’ and you measure its velocity to be v. These two velocities, v’ and v, are different vectors because they are measured in different coordinate frames. However, they are related by the following equation:

22. EXAMPLE:A boat is to cross directly across a river. The boat is able to do 15 knots in still water and the river current is 5 knots. ( A knot is 1 nautical mile per hour, a nautical mile is 6080 ft). (a) At what angle upstream must the boat be pointed? (b) What will the boat’s speed be relative to the ground?

23. ANSWER:A boat is to cross directly across a river. The boat is able to do 15 knots in still water and the river current is 5 knots. ( A knot is 1 nautical mile per hour, a nautical mile is 6080 ft). (a) At what angle upstream must the boat be pointed? (b) What will the boat’s speed be relative to the ground? u = v water v’=v boat, water v=v boat, gnd 5 kn 15 kn This vector problem can be solved using the fact that these vectors form a right- triangle.