two-dimensional motion and vectors chapter 3in.sjprep.org/science/collins/physics/vectors again...
TRANSCRIPT
VECTORS - AGAINVECTORS AGAIN
Scalar- a quantity that has magnitude but no Scalar a quantity that has magnitude but no directionVector physical quantity that has both Vector- physical quantity that has both magnitude and directionR lt t th f d h ddi g t Resultant – the answer found when adding two or more vectorsEquilibrant- equal in magnitude to the resultant in the opposite direction (180˚)
VECTORS – AGAIN CONTVECTORS AGAIN, CONT.
Vectors can be added graphically Draw the vectors to scale
I.e. 1cm=2m
and in the correct directionAll vectors are drawn tip to tail All vectors are drawn tip to tail. The resultant is drawn from the tail of the fi h i f h lfirst to the tip of the last(I won’t make you do this again)But you will have to add vectors algebraically
SOME ADDITIONAL NOTES ON VECTORSSOME ADDITIONAL NOTES ON VECTORS
Vectors can be added in any orderVectors can be added in any orderWhen adding two or more vectors, the sum is independent of the order of the additionindependent of the order of the addition
See diagram on the bottom of pg 84
T bt t t dd it itTo subtract a vector you add its opposite.
COORDINATE SYSTEMCOORDINATE SYSTEMWe use compass directionsdirectionsAnd angles from a reference directionreference directionEx: 300m/s 30˚ North of eastNorth of eastEast is the reference directiondirectionOr: 300m/s 60˚ east of northeast of north
DETERMINING RESULTANT MAGNITUDE AND DIRECTION
Pythagorean theorem for right trianglesPythagorean theorem for right triangles222 bac +=
Tangent, sine, and cosine
hypotenuseopposite
=θsin
ihypotenuseadjacent
=θcos
adjacentopposite
=θtan
EXAMPLEEXAMPLEAn archaeologist climbs the Great Pyramid in Giza,
Egypt The pyramid’s height is 136m and its width Egypt. The pyramid s height is 136m and its width is 2.30x102 m. What is the magnitude and direction of the displacement of the archaeologist direction of the displacement of the archaeologist after he has climbed from the bottom of the pyramid to the top?
y Δy= 136 mΔx=1/2 width=115md=?
Δy=136md
θ=?
xΔx=115m
θ
RESOLVING VECTORS INTO COMPONENTSRESOLVING VECTORS INTO COMPONENTS
In the pyramid example, the horizontal and In the pyramid example, the horizontal and vertical parts that add up to give the total displacement are call componentsdisplacement are call componentsAll vectors can be broken up into x and y components using right triangle mathcomponents using right triangle math
Not all motion is along a straight path in one Not all motion is along a straight path in one dimensionMost motion involves two dimensionsMost motion involves two dimensionsVectors and vector components help to simplify th 2 d tithe 2-d motion
HORIZONTAL PROJECTILESHORIZONTAL PROJECTILES
Motion in a plane means motion in two Motion in a plane means motion in two directionsWe must consider not only the horizontal (x) We must consider not only the horizontal (x) direction, but also the vertical (y)W th ti d b f b t We use the same equations used before- but now have equations for both x and y
THE VERTICAL (Y) DIRECTIONTHE VERTICAL (Y) DIRECTIONAcceleration is constant “g”=-9.81 constant g 9.81 m/s2
Initial velocity (viy) is ZERO
yZEROFinal velocity (vfy) is given or calculatedgiven or calculatedTime (t) to strike the ground is given or can be calculatedHeight of release (dy) is given or can be given or can be calculated
THE VERTICAL (Y) EQUATIONSTHE VERTICAL (Y) EQUATIONS
Consider this equation from before but for the Consider this equation from before but for the y-direction
21 tatvd +2
tatvd yiyy +=
Remember that viy = 0 and ay=g, therefore-This becomes a simple free-fall problem!dy=1/2gt2
THE HORIZONTAL (X) DIRECTIONTHE HORIZONTAL (X) DIRECTION
Velocity is considered CONSTANT in the x-Velocity is considered CONSTANT in the xdirection! vix=vfx
This means that acceleration is ZEROThis means that acceleration is ZEROHorizontal distance (dx) traveled (range) is given
b l l t dor can be calculatedTime (t) must be calculated from the vertical direction equations, then substituted in
THE HORIZONTAL (X) EQUATIONSTHE HORIZONTAL (X) EQUATIONS
Consider this equation but for the x-directionConsider this equation but for the x direction
21 tatvd +2
tatvd xixx +=
Remember that vx is constant, and ax =0The equation simplifies to this-The equation simplifies to thisDx=vixt
AN EXAMPLEAN EXAMPLE
An airplane traveling at 100.m/s drops a An airplane traveling at 100.m/s drops a package from a height of 3000. meters. Calculate the time it takes to reach the ground Calculate the time it takes to reach the ground and how far in front of the target must the package be dropped?package be dropped?
THE SOLUTION!THE SOLUTION!
Given: vi =0.0m/s vi =100.m/sGiven: viy 0.0m/s vix 100.m/sdy=3000. m ay=g=-9.81m/s2
T k th d d di ti iti t Take the downward direction as positive to eliminate negative signFind: (1) time to reach the ground
(2) distance from the drop point to targetgSolution: divide the problem into two portions, vertical and horizontalvertical and horizontal
(1) VERTICAL(1) VERTICAL
At the beginning, the plane was moving At the beginning, the plane was moving horizontally, therefore – viy =0
1 2
21 tad yy =
3000.m = ½(9.81m/s2)t2
24.7 s =t
(2) HORIZONTAL(2) HORIZONTAL
The plane and package were traveling at a The plane and package were traveling at a constant speed, therefore-V =100 m/s ax=0 m/s2Vix =100.m/s ax=0 m/s2
dx=vixtdx=(100. m/s)(24.7s) dx=2470mx
The package should be released 2470m before the targetthe target