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.

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.

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.

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

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

Projectile Motion

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

EXAMPLE: The Royal Gorge Bride rises 321 m above the Arkansas River. Suppose you kick a little rock horizontally off the bridge. The rock hits the water such that the magnitude of its horizontal displacement is 45.0 m. Find the speed at which the rock was kicked.

Projectiles launched at an Angle x = vicos()t vy,f = visin()-gt vy,f2 = vi2(sin())2 - 2gy y = visin()t-1/2gt2

Upwardly Launched Projectiles projectile motion simulator lady bug simulator

EXAMPLE: A zoo keeper finds an escaped money hanging from a light pole. Aiming her tranquilizer gun a the monkey, the zoo keeper kneels 10.0 m from the light pole, which is 5 m high. The tip of her gun is 1 m above the ground. The monkey tries to trick the zoo keeper by dropping a banana, then continues to hold onto the light pole. At the moment the monkey releases the banana, the zoo keeper shoots. If the tranquilizer dart travels at 50 m/s, will the dart hit the money, banana, or neither one?

Relative Motion & Frames of Reference Observers using different frames of reference may measure different displacements or velocities for an object in motion. Example: Airplane dropping stunt dummy

EXAMPLE: A boat heading north crosses a wide river with a velocity of 10.00 km/h relative to the water. The river has a uniform velocity of 5.00 km/h due east. Determine the boat's velocity with respect to an observer on shore.

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 The earth revolves around the sun and rotates on its axis.

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

Angular Displacement: how much an object has rotated   = s change in arc length r distance from axis

Angular Displacement EXAMPLE: While riding on a carousel that is rotating clockwise, a child travels through an arc length of 11.5 m. If the child's angular displacement is 165o what is the radius of the carousel?

 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

Angular Speed EXAMPLE: A child at an ice cream parlor spins on a stool. The child turns counterclockwise with an average angular speed of 4 rad/s. In what time interval will the child's feet have an angular displacement of 8 rad?

Tangential Speed EXAMPLE: The radius of a CD in a computer is 0.0600 m. If a microbe riding on the disc's rim has a tangential speed of 1.88 m/s, what is the disc's angular speed?

 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.

Angular Acceleration EXAMPLE: a car's tire rotates at an initial angular speed of 21.5 rad/s. The driver accelerates, and after 3.5s the tire's angular speed is 28.0 rad/s. What is the tire's average angular acceleration during the 3.5 s time interval.

Tangential Acceleration EXAMPLE: A spinning ride at a carnival has an angular acceleration of 0.5 rad/s2. How far from the center is a rider who has a tangential acceleration of 3.3 m/s2?

EXAMPLE: The wheel on an upside moves through 11.0 rad in 2 s. What is the wheel's angular acceleration if its initial angular speed is 2 rad/s?

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.

Centripetal Acceleration EXAMPLE: A test car moves at a constant speed around a circular track. If the car is 48.2 m from the track's center and has a centripetal acceleration of 8.05 m/s2, what is its tangential speed?

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

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