chapter 3: motion in 2 or 3 dimensions. position & velocity vectors
TRANSCRIPT
Chapter 3: Motion in 2 or 3 Dimensions
Position & Velocity Vectors
Position Vector
To describe the motion of a particle in space, we first need to describe the position of the particle.
Position vector of a particle is a vector that goes from the origin of coordinate system to the point.
Position vector components are the Cartesian coordinates of the particle.
kzjyixr ˆˆˆ
Position & Velocity Vectors
As the particle moves through space, the path is a curve.
The change in position (the displacement) of a particle during time interval t is:
kzzjyyixxrrr ˆ)(ˆ)(ˆ)( 12121212
Average velocity vector during this time interval is the displacement divided by the time interval:
t
r
tt
rrvav
12
12
Position & Velocity Vectors
Magnitude of the vector v at any instant is the speed v of the particle at that instant.
Direction of v at any instant is the same as the direction in which particle moves at that instant.
Instantaneous velocity vector is the limit of the average velocity as the time interval approaches zero, and equals the instantaneous rate of change of position with time:
dt
rd
t
rv
t
0
lim
Position & Velocity Vectors
As t0, P1 and P2 move closer and in this limit vector r becomes tangent to the curve.
Direction of r in the limit is the same as direction of instantaneous velocity v.
At every point along the path, instantaneous velocity vector v is the tangent to the path at that point.
dt
dxvx
dt
dyvy
dt
dzvz
Position & Velocity Vectors
Components of instantaneous velocity vector v :
dt
dxvx
dt
dyvy
dt
dzvz
kdt
dzj
dt
dyi
dt
dx
dt
rdv ˆˆˆ
222zyx vvvvv
Magnitude of vector v by Pythagorean theorem:
Acceleration Vector
Acceleration Vector
Acceleration of a particle moving in space describes how the velocity of particle changes.
Average acceleration is a vector change in velocity divided by the time interval:
t
v
tt
vvaav
12
12
Acceleration Vector
Instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero, and equals the instantaneous rate of change of velocity with time:
dt
vd
t
va
t
0
limdt
dva xx
dt
dva yy
dt
dva zz
Acceleration Vector
In terms of unit vectors: kdt
dvj
dt
dvi
dt
dva zyx ˆˆˆ
2
2
dt
xdax
2
2
dt
yday
2
2
dt
zdaz
kdt
zdj
dt
ydi
dt
xda ˆˆˆ
2
2
2
2
2
2
Acceleration Vector
Components of acceleration:
Acceleration vector can be resolved into a component parallel to the path (and velocity), and a component perpendicular to the path.
Acceleration Vector
Components of acceleration:
When acceleration vector is parallel to the path (and velocity), the magnitude of v increases, but its direction doesn’t change
When acceleration vector is perpendicular to the path (and velocity), the direction of v changes, but magnitude is constant
Acceleration Vector
Components of acceleration for a particle moving along a curved path:
A. Constant speed
B. Increasing speed
C. Decreasing speed
Projectile Motion
Projectile Motion
A projectile is any object that is given an initial velocity and then follows a path (trajectory) determined solely by gravity and air resistance.
The motion of a projectile will take place in a plane (so, it is 2-D motion).
For projectile motion we can analyze the x- and y-components of the motion separately.
The horizontal motion (along the x-axis) will have zero acceleration and thus have constant velocity.
The vertical motion (along the y-axis) will have constant downward acceleration of magnitude g = 9.80 m/s2.
The initial velocity components, vox and voy, can be expressed in terms of the magnitude vo and direction o of the initial velocity.
Projectile Motion
Projectile Motion
We analyze projectile motion as a combination of horizontal motion with constant velocity and vertical motion with constant acceleration
tvxxvva xxxx 000 ,0
200
0
5.0
,
gttvyy
gtvvga
y
yyy
000000 sin,cos vvvv yx
Initial velocity is represented by its magnitude and direction
Projectile Motion
Trajectory of a body projected with initial velocity v0
h is maximum height of trajectory
R is horizontal range
gtvv
vv
gttvy
tvx
y
x
00
00
200
00
sin
cos
5.0)sin(
)cos(
Projectile Motion
Projectile Motion
Trajectory of a cow
A cow is launched from the top of a hill with an initial velocity vector that makes an angle of 45 degrees with the horizontal. The projectile lands at a point that is 10 m vertically below the launch point and 300 m horizontally away from the launch point.
A. Determine the time the cow was in the air. B. Determine the initial speed of the cow.
Projectile Motion
2
022
00 cos2)(tan x
v
gxy
Initial speed of the cow
Projectile Motion
Flight time of the cow
200
00
5.0)sin(
)cos(
gttvy
tvx
Motion in a Circle
Motion in a Circle
When a particle moves along a curve, direction of its velocity vector changes.
Particle must have component of acceleration to the curved path even if the speed is constant.
Motion in a circle is a special case of motion along a curved path.
Uniform circular motion - when a particle moves with constant speed
Non-uniform circular motion - if the speed of a particle varies.
Uniform Circular Motion
No component of acceleration parallel (tangent) to the path. Otherwise, speed would change.
Non-zero component of acceleration is perpendicular to the path.
Uniform Circular Motion
A particle that is undergoing motion in such a manner that its direction is changing is experiencing a radial acceleration that has magnitude equal to the square of its velocity divided by the instantaneous radius of curvature of its motion. The direction of this radial, or centripetal, acceleration is toward the center of circular path of particle's motion.
R
s
v
v
1
sR
vv 1
t
s
R
v
t
vaav
1
t
s
R
v
t
s
R
va
tt
0
11
0limlim
R
varad
2
Uniform Circular Motion
In uniform circular motion, the magnitude a of instantaneous acceleration is equal to the square of the speed v divided by the radius R of the circle.
Its direction is to v and inward along the radius.
Centripetal “seeking the center” (Greek)
Uniform Circular Motion
Period of the motion T the time of one revolution, or one complete trip around the circle.
In time T, particle travels the distance 2Rof the circle, so its speed can be expressed as T
Rv
2
R
varad
2
2
4
T
Rarad
Motion in a Circle: Example
Centripetal acceleration on a curved road A car has a “lateral acceleration” of 0.87g, which is
(0.87)/(9.8m/s2)=8.5m/s2. This represents the maximum centripetal acceleration that can be attained without skidding out of the circular path. If the car is traveling at a constant speed 40m/s (~89mi/h, or 144km/h), what is the max radius of curve it can negotiate?
R
varad
2
IDENTIFY and SET UP Car travels along a curve, speed is constant apply equation of
circular motion to find the target variable R.
EXECUTE We know arad and v, so
we can find R: m
a
vR
sm
sm
rad
1905.8
)40(
2
22
Non-Uniform Circular Motion
R
varad
2
dt
vda
tan
An object that is undergoing non-uniform circular motion, or motion where the magnitude and the direction of the velocity is changing, will experience an acceleration that can be described by two components:
A radial or centripetal acceleration equal to the square of speed divided by radius of curvature of motion directed toward the center of curvature of the motion, and
Tangential component of acceleration that is equal to the rate of change of the particle's speed and is directed either parallel (in the case of speeding up) or anti-parallel (in the case of slowing down) to the particle's velocity.
Relative Velocity
Relative Velocity
The velocity seen by particular observer is called relative to that observer, or relative velocity.
Relative Velocity in 1-D
Woman walks with a velocity of 1.0m/s along the aisle of a train that is moving with a velocity of 3.0m/s. What is the woman’s velocity?
For passenger sitting in a train: 1.0m/s
For bicyclist standing: 1.0m/s + 3.0m/s = 4.0m/s
t
r
tt
rrvav
12
12
Frame of reference is a coordinate system + time scale
Relative Velocity in 1-D
Cyclist: frame of reference A
Moving train: frame of reference B
In 1-D motion, position of P relative to frame of reference A is given by distance XP/A
Position of P relative to frame of reference B is given by distance XP/B
ABBPAP xxx ///
Distance from origin A to origin B is given by XB/A
Relative Velocity in 1-D
Velocity VP/A of P relative to frame A is the derivative of XP/A with respect to time
dt
dx
dt
dx
dt
dx ABBPAP /// ABBPAP vvv ///
sm
ABsm
BP vv 0.30.1 //
sm
sm
sm
APv 0.40.30.1/