PSU PHYS 250 - Motion in Two Dimensions

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Phys 250 Ch3 p1Motion in Two DimensionsGalileo’s study of motion included exploration of motion in two dimensions (animations)Vectors are used to describe motionVectors have magnitude and directionScalars are simply numbers (i.e. magnitude only)Vectors Scalarsdisplacement distancevelocity speedaccelerationforceVectors are denote by bold face or arrowsThe magnitude of a vector is denoted by plain text or vertical barsVectors can be graphically represented by arrowsnote direction and magnitudeVorVVorVPhys 250 Ch3 p2Vector Addition: Graphical Method of R = A + B•Shift B parallel to itself until its tail is at the head of A, retaining its original length and direction.•Draw R (the resultant) from the tail of A to the head of B.AB+ =AB=Rthe order of addition of several vectors does not matterACBDABCDDBACPhys 250 Ch3 p3the order of addition of several vectors does not matterACBDABCDDBACPhys 250 Ch3 p4Vector Subtraction: the negative of a vector points in the opposite direction, but retains its size (magnitude)• A B = A +( )AB =RAB=ABPhys 250 Ch3 p5Resolving a Vectorreplacing a vector with two or more (mutually perpendicular) vectors => componentsdirections of components determined by coordinates or geometry.AAyAxExamples:horizontal and verticalNorth-South and East-WestAyAxA222adjacentoppositetanhypotenuseadjacentcoshypotenuseoppositesinAAAAAAACAyxxyxyRemember basic trig: SOH CAH TOAPhys 250 Ch3 p6Example: Swimming at an angle of 27º from the horizontal, an angelfish ha a velocity v with magnitude 25 cm/s. Find the horizontal and vertical components of v.Phys 250 Ch3 p7Vector Addition by componentsR = A + B + CResolve vectors into components(Ax, Ay etc. )Add like componentsAx + Bx + Cx = Rx Ay + By + Cy = RyThe magnitude and direction of the resultant R can be determined from its components. Strategy:1. Draw a sketch and choose a coordinate system. Use graphical method to estimate result2. Resolve all vectors into components3. Add all x-components to get the resultant x-component. Add all y-components to get the resultant y-component.4. Determine the magnitude and direction of the resultant vector, as needed.Phys 250 Ch3 p8Example: Vector A has length 14 cm at 60º with respect to the +x-axis, and vector B has length 20 at 20º with respect to the +x-axis. What is the resultant of A+BPhys 250 Ch3 p9Relative VelocityTwo frames of reference, one “moving relative to the other”A school bus is traveling at 20m/s relative to the crossing guard. A boy on the bus rolls a ball from the back of the bus to the front with a speed of 5m/s relative to the boy.How fast does the ball go, elative to the crossing guard?How does this change if the ball is rolled from the front to the back of the bus?How does this change if the ball is rolled from the front to the back of the bus?Relative VelocityVAC = VAB + VBC velocity of A relative to C equals velocity of A relative to B plusvelocity of B relative to CPhys 250 Ch3 p10Example: A person can row a boat 5.00 km/hr in still water tries to cross a river whose current is 3.00 km/hr. The boat is pointed straight across the river, but it is carried downstream by the river as the rower rows across.What is the velocity of the boat relative to land?How far down stream does the boat land on the opposite shore if the river is 200 km wide?A small airplane with an airspeed of 200 km/hr is flown directly north by a novice pilot from Columbia to Charlotte. The wind is blowing from northwest to sought east at 28 km/hr. What is the plane’s resultant speed and direction relative to the ground?Phys 250 Ch3 p11Kinematics in Two DimensionsRates of change:average velocityChange in vector  change in components!instantaneous velocity and accelerationvectors!3-D kinematics:but look at components...trvttttvarv00limlim2222yxyxaaavvv 200021tttavrravv200020002121tatvyytavvtatvxxtavvyyyyyxxxxxPhys 250 Ch3 p12Example: A particle is confined to move in a horizontal plane. It starts at the origin at t=0. The particle has an initial velocity of 10 cm/s directed along the +x axis and an acceleration of 2 cm/s2 in +y direction. Compute the particle’s position at t=1,2,3,4,5 sWhat is the particles velocity at t=1,3,5s?Sketch the path of the particle by plotting the position at the indicated times.Phys 250 Ch3 p13Projectile Motionacceleration of gravity directed vertically downno horizontal acceleration(taking up as +y direction)200000021gttvyytvxxgtvvvvyxyyxxPhys 250 Ch3 p14Example: A ball is thrown horizontally from the leaning tower of Pisa with a velocity of 22 m/s. If the ball is thrown from a height of 49 m above the ground, where will the ball hit the ground?discussion: monkey and the dart gunPhys 250 Ch3 p15Projectile RangeRv0)45ºat (Rrange) same with the(angles 902sincossin2sin2cos22impact at 2120max12202000000000200gvgvRgvgvvgvvtvxxRgvtygttvyyyxxyyPhys 250 Ch3 p16Example: An arrow leaves a bow at 30 m/s.What is its maximum range?At what two angles could the archer point the arrow for a target 70 m


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PSU PHYS 250 - Motion in Two Dimensions

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