Sec 3-1 Notes      Introduction to vectors

 

scalar – a physical quantity that has only                               magnitude but no direction

 

vector – a physical quantity that has both                               magnitude and a direction

 

examples:        scalar              vector

                        speed              velocity

                        distance          displacement

                        mass               acceleration

                        time                 Force

 

Vectors are represented in BOLDFACE in textbooks or with arrows above them in script.

 

      v, a, F        or in script       v, a, F

 

- use arrows in diagrams

 

- the length of the arrow represents the magnitude of the vector

 

- point the arrow in the correct direction

Addition of vectors graphically

·      place vectors tail to head

·      choose a scale appropriate for the distances

·      use a protractor to get the correct angle

 

example:  40 m N then 30 m E

 

 

 

 

example:  20m E

                  50m @ 20° N of E

                  80m @ 45° W of N

                  20m S

                  10m @ 30° S of E

 

There are multiple ways to communicate angles.

 

angle notation – only an angle is given

      - the angle is between 0° and 360°

      - measure this angle from the +x-axis

 

bearing notation – an angle and two directions                                        are given

 

      Method 1                     Method 2

30° W of N                              N 30° W

 

20° S of W                        W 20° S

 

examples:  write in two other ways

 

4cm N 25° E               

 

8cm S 65° E

 

example:  A plane travels at 250 km/h N with a wind of 75 km/h southwest.

Properties of Vectors

 

·      In a diagram, vectors may be moved parallel to themselves.

 

·      Order does not effect addition of vectors.  (Vector addition is commutative.)

 

·      To subtract vectors, add the opposite.

 

·      Vectors can be multiplied or divided by scalars to yield shorter or longer vectors.

Sec 3-2 notes       Vector Operations

- axis orientation – we usually use this form

- you may change but beware of complications

 

 

We can also manipulate vectors algebraically.

 

For perpendicular vectors

      - use Pythagorean theorem

     

      - use SOHCAHTOA    - trig functions

 

            sin q = opposite side / hypotenuse

     

            cos q = adjacent side / hypotenuse

     

            tan q = opposite side / adjacent side

example:  A plane travel 1540 km east and then 1160 km north.  What is its displacement?

 

 

 

 

 

 

 

 

 

 

 

 

example:  A camper walks 4.5 km NE and 4.5 km NW.  What is her displacement?

vector resolution – breaking a vector down into its x and y components

- these are also called projections of the vector onto the x and y axes

 

example:  v = 45 m/s @ 25°

      Find the x and y components of this vector.

      vx =

     

 

      vy =

 

 

For Non-perpendicular vectors

      - use vector resolution

      - use the Law of Sines and Law of Cosines

 

Steps for vector resolution

·      break each vector into its x and y components – WATCH SIGNS

·      total the x components

·      total the y components

·      draw a new right triangle

·      solve

example:  1st  - 20 km @ 20° N of E

                  2nd – 50 km @ 40° S of E

Sec 3-3 notes                   Projectile Motion

 

projectile motion – free fall with an initial                                           horizontal velocity

 

projectiles – objects that are thrown or launched into the air and are subject to gravity

 

trajectory – the path an object travels

- projectiles have a parabolic trajectory

 

To solve projectile motion problems, work the vertical and horizontal components separately.

 

For projectiles launched horizontally with vi,

      vix = vi         and      viy = 0 m/s

 

 

For projectiles launched at an angle q with vi,

      vxi = vi cosq and      vyi = vi sinq

horizontal equations                vertical equations

            (x)                                            (y)             

 

      Dx = vxiDt                            vyf = vyi + at

                                                Dy = vyit + ½ at2

                                                vyf2 = vyi2 + 2aDy

 

horizontal equations determine range

      – how far away the object gets

 

vertical equations determine height

 

TIME is the only common element.

 

example:  A girl stands on a 50m building and throws a ball horizontally at 12 m/s east.

      a.  How long is the ball in the air?

      b.  What are its horizontal and vertical           velocities when it strikes the ground?

      c.  What is its resultant velocity when it    strikes the ground?

      d.  How far is it from the base of the   building?

 

example:  A rock is thrown horizontally off of a cliff that is 1.2m high and lands 3m away from the base of the cliff.  What was its initial velocity?

example:  A golfer practices driving balls off a cliff and into water below.  The cliff is 15m from the water. If the golf ball is launched at 51 m/s at an angle of 15°, how far does the ball travel horizontally before hitting the water?  What was the maximum height of the ball?

Sec. 3-4 Relative Motion

 

*Velocity measurements differ in different frames of reference.

 

example: You are traveling at 45 mph and a car traveling at 57 mph passes you.  What is the speed of that car relative to you?

 

 

 

Consider using subscript (abbreviated) to help you with these problems.

 

      vye = velocity of you with respect to the earth

 

      vce = velocity of car with respect to the earth

 

      vcy = velocity of car with respect to you

 

arrange these so that the subscripts start with c and end with y.

 

      vcy = vce + vey               note that vey = -vye

 

so vcy = 57 + -45

 

example:  A plane flies northeast at an airspeed of 563.0 km/h.  A 48.0 km/h wind is blowing to the southeast.  What is the plane’s velocity relative to the ground?