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TRANSCRIPT
Raymond A. Serway Chris Vuille
Chapter Three Vectors and
Two-‐Dimensional Mo=on
Vectors and Mo=on
• In one-‐dimensional mo=on, vectors were used to a limited extent
• For more complex mo=on, manipula=ng vectors will be more important
Introduc=on
Vector vs. Scalar Review
• All physical quan==es encountered in this text will be either a scalar or a vector
• A vector quan=ty has both magnitude (size) and direc=on
• A scalar is completely specified by only a magnitude (size)
Sec=on 3.1
Vector Nota=on
• When handwriNen, use an arrow: • When printed, will be in bold print with an arrow:
• When dealing with just the magnitude of a vector in print, an italic leNer will be used: A – Italics will also be used to represent scalars
Sec=on 3.1
Proper=es of Vectors
• Equality of Two Vectors – Two vectors are equal if they have the same magnitude and the same direc=on
• Movement of vectors in a diagram – Any vector can be moved parallel to itself without being affected
Sec=on 3.1
Adding Vectors
• When adding vectors, their direc=ons must be taken into account
• Units must be the same • Geometric Methods – Use scale drawings
• Algebraic Methods • The resultant vector (sum) is denoted as
Sec=on 3.1
Adding Vectors Geometrically (Triangle or Polygon Method)
• Choose a scale • Draw the first vector with the appropriate length and in the direc=on specified, with respect to a coordinate system
• Draw the next vector using the same scale with the appropriate length and in the direc=on specified, with respect to a coordinate system whose origin is the end of the first vector and parallel to the ordinate system used for the first vector
Sec=on 3.1
Graphically Adding Vectors, cont. • Con=nue drawing the
vectors “=p-‐to-‐tail” • The resultant is drawn from
the origin of the first vector to the end of the last vector
• Measure the length of the resultant and its angle – Use the scale factor to
convert length to actual magnitude
• This method is called the triangle method
Sec=on 3.1
Notes about Vector Addi=on
• Vectors obey the Commuta/ve Law of Addi/on – The order in which the vectors are added doesn’t affect the result
–
Sec=on 3.1
Graphically Adding Vectors, cont. • When you have many
vectors, just keep repea=ng the “=p-‐to-‐tail” process un=l all are included
• The resultant is s=ll drawn from the origin of the first vector to the end of the last vector
Sec=on 3.1
More Proper=es of Vectors
• Nega=ve Vectors – The nega=ve of the vector is defined as the vector that gives zero when added to the original vector
– Two vectors are nega/ve if they have the same magnitude but are 180° apart (opposite direc=ons)
–
Sec=on 3.1
Vector Subtrac=on
• Special case of vector addi=on – Add the nega=ve of the subtracted vector
• • Con=nue with standard vector addi=on procedure
Sec=on 3.1
Mul=plying or Dividing a Vector by a Scalar
• The result of the mul=plica=on or division is a vector • The magnitude of the vector is mul=plied or divided by the scalar
• If the scalar is posi=ve, the direc=on of the result is the same as of the original vector
• If the scalar is nega=ve, the direc=on of the result is opposite that of the original vector
Sec=on 3.1
Components of a Vector
• It is useful to use rectangular components to add vectors – These are the projec=ons of the vector along the x-‐ and y-‐axes
Sec=on 3.2
Components of a Vector, cont.
• The x-‐component of a vector is the projec=on along the x-‐axis –
• The y-‐component of a vector is the projec=on along the y-‐axis –
• Then,
Sec=on 3.2
More About Components of a Vector
• The previous equa=ons are valid only if Θ is measured with respect to the x-‐axis
• The components can be posi=ve or nega=ve and will have the same units as the original vector
Sec=on 3.2
More About Components, cont.
• The components are the legs of the right triangle whose hypotenuse is
– May s=ll have to find θ with respect to the posi=ve x-‐axis – The value will be correct only if the angle lies in the first or fourth quadrant
– In the second or third quadrant, add 180°
Sec=on 3.2
Other Coordinate Systems • It may be convenient to use
a coordinate system other than horizontal and ver=cal
• Choose axes that are perpendicular to each other
• Adjust the components accordingly
Sec=on 3.2
Adding Vectors Algebraically
• Choose a coordinate system and sketch the vectors
• Find the x-‐ and y-‐components of all the vectors
• Add all the x-‐components – This gives Rx:
Sec=on 3.2
Adding Vectors Algebraically, cont.
• Add all the y-‐components – This gives Ry:
• Use the Pythagorean Theorem to find the magnitude of the resultant:
• Use the inverse tangent func=on to find the direc=on of R:
Sec=on 3.2
Mo=on in Two Dimensions
• Using + or – signs is not always sufficient to fully describe mo=on in more than one dimension – Vectors can be used to more fully describe mo=on
• S=ll interested in displacement, velocity, and accelera=on
Sec=on 3.3
Displacement • The posi=on of an object is
described by its posi=on vector,
• The displacement of the object is defined as the change in its posi8on – – SI unit: meter (m)
Sec=on 3.3
Velocity
• The average velocity is the ra=o of the displacement to the =me interval for the displacement
• The instantaneous velocity is the limit of the average velocity as Δt approaches zero – The direc=on of the instantaneous velocity is along a line that is tangent to the path of the par=cle and in the direc=on of mo=on
• SI unit: meter per second (m/s)
Sec=on 3.3
Accelera=on
• The average accelera=on is defined as the rate at which the velocity changes
• The instantaneous accelera=on is the limit of the average accelera=on as Δt approaches zero
• SI unit: meter per second squared (m/s²)
Sec=on 3.3
Unit Summary (SI)
• Displacement – m
• Average velocity and instantaneous velocity – m/s
• Average accelera=on and instantaneous accelera=on – m/s2
Sec=on 3.3
Ways an Object Might Accelerate
• The magnitude of the velocity (the speed) may change with =me
• The direc=on of the velocity may change with =me – Even though the magnitude is constant
• Both the magnitude and the direc=on may change with =me
Projec=le Mo=on
• An object may move in both the x and y direc=ons simultaneously – It moves in two dimensions
• The form of two dimensional mo=on we will deal with is an important special case called projec/le mo/on
Sec=on 3.4
Assump=ons of Projec=le Mo=on
• We may ignore air fric=on • We may ignore the rota=on of the earth • With these assump=ons, an object in projec=le mo=on will follow a parabolic path
Sec=on 3.4
Rules of Projec=le Mo=on
• The x-‐ and y-‐direc=ons of mo=on are completely independent of each other
• The x-‐direc=on is uniform mo=on – ax = 0
• The y-‐direc=on is free fall – ay = -‐g
• The ini=al velocity can be broken down into its x-‐ and y-‐components –
Sec=on 3.4
Projec=le Mo=on
Sec=on 3.4
Projec=le Mo=on at Various Ini=al Angles
• Complementary values of the ini=al angle result in the same range – The heights will be different
• The maximum range occurs at a projec=on angle of 45o
Sec=on 3.4
Some Details About the Rules
• x-‐direc=on – ax = 0 – – x = voxt
• This is the only opera=ve equa=on in the x-‐direc=on since there is uniform velocity in that direc=on
Sec=on 3.4
More Details About the Rules
• y-‐direc=on – – Free fall problem
• a = -‐g – Take the posi=ve direc=on as upward – Uniformly accelerated mo=on, so the mo=on equa=ons all hold
Sec=on 3.4
Velocity of the Projec=le
• The velocity of the projec=le at any point of its mo=on is the vector sum of its x and y components at that point
– Remember to be careful about the angle’s quadrant
Sec=on 3.4
Projec=le Mo=on Summary
• Provided air resistance is negligible, the horizontal component of the velocity remains constant – Since ax = 0
• The ver=cal component of the accelera=on is equal to the free fall accelera=on –g – The accelera=on in the y-‐direc=on is not zero at the top of the projec=le’s trajectory
Sec=on 3.4
Projec=le Mo=on Summary, cont
• The ver=cal component of the velocity vy and the displacement in the y-‐direc=on are iden=cal to those of a freely falling body
• Projec=le mo=on can be described as a superposi=on of two independent mo=ons in the x-‐ and y-‐direc=ons
Sec=on 3.4
Problem-‐Solving Strategy
• Select a coordinate system and sketch the path of the projec=le – Include ini=al and final posi=ons, veloci=es, and accelera=ons
• Resolve the ini=al velocity into x-‐ and y-‐components
• Treat the horizontal and ver=cal mo=ons independently
Sec=on 3.4
Problem-‐Solving Strategy, cont
• Follow the techniques for solving problems with constant velocity to analyze the horizontal mo=on of the projec=le
• Follow the techniques for solving problems with constant accelera=on to analyze the ver=cal mo=on of the projec=le
Sec=on 3.4
Some Varia=ons of Projec=le Mo=on
• An object may be fired horizontally
• The ini=al velocity is all in the x-‐direc=on – vo = vx and vy = 0
• All the general rules of projec=le mo=on apply
Sec=on 3.4
Non-‐Symmetrical Projec=le Mo=on • Follow the general rules for
projec=le mo=on • Break the y-‐direc=on into
parts – up and down – symmetrical back to ini=al
height and then the rest of the height
Sec=on 3.4
Special Equa=ons
• The mo=on equa=ons can be combined algebraically and solved for the range and maximum height
Sec=on 3.4
Rela=ve Velocity
• Rela=ve velocity is about rela=ng the measurements of two different observers
• It may be useful to use a moving frame of reference instead of a sta=onary one
• It is important to specify the frame of reference, since the mo=on may be different in different frames of reference
• There are no specific equa=ons to learn to solve rela=ve velocity problems
Sec=on 3.5
Rela=ve Velocity Nota=on
• The paNern of subscripts can be useful in solving rela=ve velocity problems
• Assume the following nota=on: – E is an observer, sta=onary with respect to the earth
– A and B are two moving cars
Sec=on 3.5
Rela=ve Posi=on Equa=ons
• is the posi=on of car A as measured by E • is the posi=on of car B as measured by E • is the posi=on of car A as measured by car B
•
Sec=on 3.5
Rela=ve Posi=on
• The posi=on of car A rela=ve to car B is given by the vector subtrac=on equa=on
Sec=on 3.5
Rela=ve Velocity Equa=ons
• The rate of change of the displacements gives the rela=onship for the veloci=es
Sec=on 3.5
Problem-‐Solving Strategy: Rela=ve Velocity
• Label all the objects with a descrip=ve leNer • Look for phrases such as “velocity of A rela=ve to B” – Write the velocity variables with appropriate nota=on
– If there is something not explicitly noted as being rela=ve to something else, it is probably rela=ve to the earth
Sec=on 3.5
Problem-‐Solving Strategy: Rela=ve Velocity, cont
• Take the veloci=es and put them into an equa=on – Keep the subscripts in an order analogous to the standard equa=on
• Solve for the unknown(s)
Sec=on 3.5
Rela=ve Velocity, Example
• Need veloci=es – Boat rela=ve to river – River rela=ve to the Earth
– Boat with respect to the Earth (observer)
• Equa=on –