Motion in 2 dimensions

Vectors vs. Scalars Scalar- a quantity described by magnitude only. –Given by numbers and units only. –Ex. Distance, speed and mass Vector – a quantity described by both magnitude and direction –Numbers, units and direction (either words or angles –Ex. Displacement, velocity, acceleration and force

vectors Drawn with an arrow Length indicates magnitude Direction pointed Written in text either –As a boldfaced letter –Or as a letter with an arrow on top. 35 m East

Vector subtraction Scalar multiplication Vectors may be subtracted by adding the opposite (or negative) of the second vector. V 2 – V 1 = V 2 + (– V 1 ) –This means that V 1 is facing the opposite direction it was originally heading Multiplying a vector by a scalar simply magnifies its length if V= 3 m then 3V = 9 m

Vector addition Vectors may be added two ways: –One by graphing Vectors are drawn so that they remain in the same orientation but are placed tip to tail. Or by the parallelogram method –One by adding the components of each and doing the Pythagorean theorem. The resultant vector is the outcome of adding 2 vectors together.

Parallelogram method: In the parallelogram method for vector addition, the vectors are translated, (i.e., moved) to a common origin and the parallelogram constructed as follows: The resultant R is the diagonal of the parallelogram drawn from the common origin.

R = resultant vector Pythagorean theorem = R 2 R = 11.2 Km Θ = tan -1 (5/10) =26.6° west of north So sum = ° west of north

Component method – make a right triangle out of the individual vector

components The tip of the x-component vector is directly below the tip of the original vector.

components,.

When adding 2 vectors by the component method Find the x and y components of each vector Organize in a table Then add x component to x component and y component to y component, VectorX componentY component A B ResultantX r YrYr

Use negative x components when vector is pointed to the left Use Negative y components pointed down Use the Pythagorean formula to find the length of the resulting vector. Find the angle using the Tan -1 (Y r /X r )

vector addition applet simulator

Example 1 What is the vertical component of a 33 m vector that is at a 76° angle with the x axis? y –comp y = sin y = 33 sin 76 = 32 m 76° 33 m

Example 2 A plane is heading to a destination 1750 due north at 175 km/hr in a westward wind blowing 25 km/hr. At what angle should the plane be oriented so that it reaches its destination? The wind will push the plane off course to the west by an angle of tan -1 (25/175) = 8° west of north so the plane needs to head 8° east of north. Wind =25 km/hr Plane = 175 km/hr

Example 3 Using the same plane in example 2, what would the magnitude of the resultant vector be? –Since north and west, don’t need components, just use the pythagorean theorem = R 2 R = 177 km/hr

example 4 What vector represents the displacement of a person who walks 15 km at 45° south of east then 30 km due west? R = √( -19.4) 2 + (-10.6) 2 R = 22.1 km Θ = tan -1 (-10.6/-19.4) Θ = 29° south of west 15 km 30 km VectorX – comp Y -comp 15 km km-300 resultant °