Two Dimensional Motion and Vectors

Two Dimensional Motion and Vectors http://www.youtube.com/watch?v=Phl2d4jeN90

Scalar--a physical quantity that has only a magnitude but no direction distance speed mass volume work energy power Vector--a physical quantity that has both a magnitude and a direction displacement velocity acceleration force momentum

Vector diagrams are diagrams that depict the direction and relative magnitude of a vector quantity by a vector arrow.

Vector Addition A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant)

1. Vectors can be moved parallel to themselves in a  diagram We can draw a given vector anywhere in the diagram as long as the vector is parallel to its previous alignment and still points in the same direction. Thus, you can draw one vector with its tail starting at the tip of the other as long as the size and direction of each vector do not change.

1. Vectors can be moved parallel to themselves in a  diagram 4. Multiplying or dividing vectors by scalars  results in vectors 2. Vectors can be added in any order. 3. To subtract a vector add its opposite.

2. Vectors can be added in any order. When two or more vectors are added, the sum is independent of the order of addition. The vector sum of two or more vectors is the same regardless of the order in which the vectors are added, provided the magnitude and direction of each vector remain the same. http://www.physicsclassroom.com/mmedia/vectors/ao.cfm

3. To subtract a vector add its opposite.

4. Multiplying or dividing vectors by scalars  results in vectors EXAMPLE: If a cab driver obeys a customer who tells him to go twice as fast, the cab's original velocity vector vcab, is multiplied by the scalar number 2. The result, 2vcab, is a vector with twice the original vector pointing in the same direction. If the cab driver is told to go twice as fast in the opposite direction, it is multiplied by the scalar -2 , two times the initial velocity but in the opposite direction.

Vectors can be added graphically Consider a student waling to school. The student walks 1500 m to a friend's house, then 1600 m to the school. School Friend's House Home Resultant--a vector representing the  sum of two or more vectors

A vector is a quantity which has both magnitude and direction. In order for such descriptions of vector quantities to be useful, it is important that everyone agree upon how the direction of an object is described. The convention upon which we can all agree is sometimes referred to as the CCW convention - counterclockwise convention. Using this convention, we can describe the direction of any vector in terms of its counterclockwise angle of rotation from due east. The direction north would be at 90 degrees since a vector pointing east would have to be rotated 90 degrees in the counterclockwise direction in order to point north. The direction of west would be at 180 degrees since a vector pointing west would have to be rotated 180 degrees in the counterclockwise direction in order to point west. Further illustrations of the use of this convention are depicted by the animation below. http://www.physicsclassroom.com/mmedia/vectors/vd.cfm

Directions

Determining resultant magnitude and direction Pythagorean Theorem a2 + b2 = c2 c a b d y x

Review of Trigonometry

Resolving vectors into components (the projections of  a vector along the axes of a coordinate system)

Example Problem: An archeologist climbs the great Pyramid in Giza, Egypt. If the pyramids height is 136 m and its width is 2.3 x 102 m, what is the magnitude and direction of the archaeologist's displacement while climbing fro the bottom of the pyramid to the top? Given: y Unknown: x

Example Problem: Find the component velocities of a helicopter traveling 95 km/hr at an angle of 35o to the ground. Given: y Unknown: vx, vy v = 95 km/h 35o x

Resolve the following vectors into X and Y components. State the results as example: Vx =+4 units, Vy = -3 units. Problem 1. V = 10.0 units at 37o east of north Problem 2. V = 4.0 units at 30o south of west

y = -1/2g(t2) Vertical motion of a projectile that falls from rest Horizontal Motion of a Projectile x = vxt

Upwardly Launched Projectiles projectile motion simulator lady bug simulator

Relative Motion & Frames of Reference

Circular (Rotational) Motion motion of a body that spins about an  axis. rotation--when an object turns about an internal axis revolution--when an object turns about an external axis

radian--an angle whose arc length is  equal to its radius ( 57.3o) one revolution = 360o = 2 radians

 ave = t vt = r Angular Speed--the rate at which a body  rotates about an axis Tangential Speed--instantaneous linear  speed of an object directed along the  tangent to the object's circular path

 ave = t at = r Angular Acceleration--change in angular  speed with time Tangential Acceleration--instantaneous  linear acceleration of an object directed  along the tangent to the object's circular  path

All points on a rotating rigid object have  the same angular speed and angular  acceleration. Tangential (linear) speed and tangential  (linear) acceleration depend upon the  radius of rotation.

v2t ac = r Centripetal (center seeking) Acceleration-- acceleration directed toward the center of a  circular path Calculate the centripetal acceleration of a race car that has a constant  tangential speed of 20.0 m/s as it moves around a circular race track with  a radius of 50.0 m.

Which part of the Earth's surface has the  greatest angular speed about the Earth's  axis? Which part has the greatest  tangential (linear) speed? angular vs tangential review

A tire rotates 3.5 times during a time  interval of 0.75 s. What is the rotational  speed of the tire in revolutions per  second?

Assuming Earth is perfectly spherical,  what is the angular speed of someone  standing on the equator in  revolutions/s?

1 If you lose your grip on a rapidly  spinning merry-go-round and fall off,  in which direction will you fly?

2 A ladybug sits halfway between the axis  and the edge of a rotating turntable.  What will happen to the ladybug's  linear speed if a. the RPM rate is doubled? b. the ladybug sits at the edge? c. both a and b occur?

3 Which state in the United States has  the greatest tangential speed as Earth  rotates around its axis?

4 The speedometer in a car is driven by a  cable connected to the shaft that turns  the car's wheels. Will speedometer  readings be more or less than actual  speed when the car's wheels are replaced  with smaller ones? A taxi driver wishes to increase his fares  by adjusting the size of his tires. Should  he change to larger tires or smaller tires?

Mars is about twice as far from the  sun as is Venus. A Martian year,  which is the time it takes Mars to go  around the sun, is about three times  as long as a Venusian year. 5 A. Which of these two planets  has the greater rotational  speed in its orbit? B. Which planet has the  greater linear speed?

Launch Speed less than 8000 m/s Projectile falls to Earth Projectile launched in the absence of  gravity Launch Speed greater than 8000 m/s Projectile orbits Earth - Elliptical Path Launch Speed equal to 8000 m/s Projectile orbits Earth - Circular Path

Two Dimensional Motion and Vectors