chapter 3 vectors and two-dimensional motion. vectors vectors – physical quantities having both...
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Chapter 3
Vectors and Two-Dimensional Motion
Vectors
• Vectors – physical quantities having both magnitude and direction
• Vectors are labeled either a or
• Vector magnitude is labeled either |a| or a
• Two (or more) vectors having the same magnitude and direction are identical
a
Vector sum (resultant vector)
• Not the same as algebraic sum
• Triangle method of finding the resultant:a) Draw the vectors “head-to-tail”b) The resultant is drawn from the tail of A to the head of B
A
B
R = A + B
Addition of more than two vectors
• When you have many vectors, just keep repeating the process until all are included
• The resultant is still drawn from the tail of the first vector to the head of the last vector
Commutative law of vector addition
A + B = B + A
Associative law of vector addition
(A + B) + C = A + (B + C)
Negative vectors
Vector (- b) has the same magnitude as b but opposite direction
Vector subtraction
Special case of vector addition: A - B = A + (- B)
Multiplying a vector by a scalar
• The result of the multiplication is a vector
c A = B
• Vector magnitude of the product is multiplied by the scalar
|c| |A| = |B|
• If the scalar is positive (negative), the direction of the result is the same as (opposite to that) of the original vector
Vector components
• Component of a vector is the projection of the vector on an axis
• To find the projection – drop perpendicular lines to the axis from both ends of the vector – resolving the vector
Vector components
x
y
yx A
AAAA tan
22
inAAAA yx s cos
Unit vectors
• Unit vector:A) Has a magnitude of 1 (unity)B) Lacks both dimension and unitC) Specifies a direction
• Unit vectors in a right-handed coordinate system
Adding vectors by components
In 2D case:
jbibb
jaiaa
yx
yx
ˆˆ
ˆˆ
bar
yyy
xxx
bar
bar
Chapter 3Problem 14
A quarterback takes the ball from the line of scrimmage, runs backwards for 10.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 50.0-yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?
Position
The position of an object is described by its position vector,
kzjyixr ˆˆˆ
r
Displacement
The displacement of the object is defined as the change in its position,
if rrr
)ˆˆ()ˆˆ( jyixjyixr iiff
jyyixx ififˆ)(ˆ)(
jyixr ˆˆ
r
Velocity
• Average velocity
• Instantaneous velocity
t
jyix
t
rvavg
ˆˆ
t
rv
t
0lim
jvivv yxˆˆ
Instantaneous velocity
Vector of instantaneous velocity is always tangential to the object’s path at the object’s position
Acceleration
• Average acceleration
• Instantaneous acceleration
t
v
t
vva if
avg
t
va
t
0lim
Acceleration
• Acceleration – the rate of change of velocity (vector)
• The magnitude of the velocity (the speed) can change – tangential acceleration
• The direction of the velocity can change – radial acceleration
• Both the magnitude and the direction can change
Projectile motion
• A special case of 2D motion
• An object moves in the presence of Earth’s gravity
• We neglect the air friction and the rotation of the Earth
• As a result, the object moves in a vertical plane and follows a parabolic path
• The x and y directions of motion are treated independently
Projectile motion – X direction
• A uniform motion: ax = 0
• Initial velocity is
• Displacement in the x direction is described as
iixi vv cos
tvxx iii )cos(
Projectile motion – Y direction
• Motion with a constant acceleration: ay = – g
• Initial velocity is
• Therefore
• Displacement in the y direction is described as
iiyi vv sin
2
2
1)sin( gttvyy iii
gtvv iiy sin
Projectile motion: putting X and Y together
constvv iix cos
2
2
1)sin( gttvyy iii
gtvv iy 0sin
tvxx iii )cos(
Projectile motion: trajectory and range
ii
g
vR 2sin
2
2
2
)cos(2)(tan
iii v
gxxy
ii
g
vh 2
2
sin2
Projectile motion: trajectory and range
ii
g
vR 2sin
2
2
2
)cos(2)(tan
iii v
gxxy
ii
g
vh 2
2
sin2
Chapter 3Problem 58
A 2.00-m-tall basketball player is standing on the floor 10.0 m from the basket. If he shoots the ball at a 40.0° angle with the horizontal, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard? The height of the basket is 3.05 m.
Relative motion
• Reference frame: physical object and a coordinate system attached to it
• Reference frames can move relative to each other
• We can measure displacements, velocities, accelerations, etc. separately in different reference frames
Relative motion
• If reference frames A and B move relative to each other with a constant velocity
• Then
• Acceleration measured in both reference frames will be the same
BAPBPA vvv
BAPBPA rrr
PBPA aa
BAv
Questions?
Answers to the even-numbered problems
Chapter 3
Problem 2: (a) Approximately 484 km (b) Approximately 18.1° N of W
Answers to the even-numbered problems
Chapter 3
Problem 6: (a) Approximately 6.1 units at 113°(b) Approximately 15 units at 23°
Answers to the even-numbered problems
Chapter 3
Problem 10: 1.31 km north, 2.81 km east
Answers to the even-numbered problems
Chapter 3
Problem 28: x = 7.23 × 103 m, y = 1.68 × 103 m