Cardinal Direction

Example: A man walks 30 m North and then 60 m West.
     a) What is the distance that he has traveled?
     b) What is his displacement ( Also state the direction he has moved from his starting point )?


Solve Graphically (more accurate if using graph paper):

     a) distance = 90 m
     b) displacement ~ 6.7 squares = 67 m, 26° + 270° = 296° bearing

Bearing: is the angle measured in degrees in a clockwise direction from the north line. 


Solve Mathematically: 

a) distance = 30 + 60 = 90 m  
b) displacement => c^2 = a^2 + b^2 = 60^2 + 30^2, c = 67.08m
     
    (Soh-Cah-Toa), Tan Theta= 30/60, Theta = 26.56°  + 270°  = 296.56° bearing 

Concurrent forces: two or more forces which act on teh same point simultanously.

• example - two people pull on ropes that are affected to a crate at the same time

Resultant: the net affect of two or more vectors acting on a point.

• solve using the parallelogram method/technique

Example: Assume there are two forces acting on a point a 50N force acting due North, and a 70N force acting due wast. The resultant would be found by first drawing parallelogram. Next the length of the diagonal between the two force is calculated. Finally direction is determined.

• Demonstration of Parallelogram

Example: One boy pull on a rope attached to a crate due North with a force of 250N. A second boy pulls on a rope attached to the same crate due West with a force of 140N. Find the resultant of the two forces acting on the crate.

     c^2 = a^2 + b^2 = 140^2 + 250^2
     c^2 = 82100, √c^2 = √82100, c = 286.53 N

 (Soh-Cah-Toa), Tan Theta= 240/140
 Theta = 1.78 = 60.67°, 

 60.67° + 270° = 330.67° bearing 

Example: An airplane is trying to fly on a bearing of 90 degrees (due east) at a velocity of 250 Km/hr. A wind force due south velocity of 130 km/hr. What is the actual velocity of the plane?

     c^2 = a^2 + b^2 => 130^2 + 250^2
     c =  281.78 km/hr
     Cos Theta = 250/281.78 = 27.50°,
     27.50° + 90° = 117.5° bearing

Example: A river is flowing due south at a velocity of 12 m/s. A boat is attempting to across this same river by traveling at a velocity of 25 m/s due west. The river is 250 m wide. 

     a) What is the velocity of the boat? (include direction)

     b) How long does it take the boat to across the river?

     c) How far down the stream is the boat by the time it reaches the opposite bank?

     a) c^2 = 25^2 + 12^2, c = 27.7 m/s
         Tan Theta = 25/12, Theta = 64.35°,
          64.35° + 180° = 244.45° bearing
     
     b) s = d/t, t = 250/25, t = 10 s

     c) d = 12 * 10 = 120 m 

Equilibrium: a state that exists when all forces cancel out.

• vector sum of all the forces equals zero
• a force which cancels out the resultant is called the equilibrant.