Post on 26-Nov-2014
transcript
Motion of a Point
Position, Velocity, and Acceleration
• Average Velocity
• Instantaneous Velocity
t
rv
rdt
rdv
Acceleration
• Average Acceleration
• Instantaneous Acceleration
t
va
rdt
rd
dt
vda
2
2
Uniform Rectilinear Motion
tvss
tvxx
vv
00
00
0
Uniformly Accelerated Rectilinear Motion
)(2
2
1
020
2
200
0
xxavv
tatvxx
tavv
Other Equations
Acceleration
dx
dvva
Acceleration as a function of velocity
v
v
t
t
dtva
dv
dtva
dv
vadt
dv
0 0)(
)(
)(
v
v
s
s
dsva
vdv
dsva
vdv
vavds
dv
vdx
dv
dt
dx
dx
dv
dt
dvva
0 0)(
)(
)(
)(
Acceleration as a function of Position
s
s
v
v o
dssavdv
dssavdv
savds
dv
vds
dv
dt
ds
ds
dv
dt
dv
sadt
dv
)(
)(
)(
)(
0
The last integral yields velocity as a function of position.
s
s
t
t
dtsv
ds
dt
dssv
0 0)(
)(
Erratic Motion
• Motion described by a piecewise function.
• Graph of s, v, a, and t.
• Motion Sensor
Position vs. Time
• Slope is velocity
• Second derivative is acceleration
Velocity vs. Time
• Slope is acceleration
• Area is displacement
Acceleration vs. Time
• Area is change in velocity.
Acceleration vs. Position
1
0
20
212
1s
s
adsvv
Velocity vs. Position
ds
dvva
Curvilinear Motion of Particles
Position Vector, Velocity, and Acceleration
rvdt
vda
rdt
rdv
Derivatives of Vector Functions
• Summation Rule
• Product Rule– Dot Product– Cross Product
Rectangular Components of Velocity and Acceleration
• Basically a summation rule.
Angular Motion
dt
ddt
d
Tangential and Normal Components
• Coordinate system
Unit vector
tt
tn
evedt
dsv
d
ede
ˆˆ
ˆˆ
nt
tt
tt
ev
edt
dv
dt
ds
ds
d
d
edve
dt
dvdt
edve
dt
dv
dt
vda
ˆˆ
ˆˆ
ˆˆ
2
2
2
2
32
1
)(
ˆcosˆsinˆ
ˆsinˆcosˆ
dxyd
dxdy
xyy
jie
jie
n
t
At a given instant in an airplane race, airplane A is flying horizontally in a straight line, and its speed is being increased at a rate of 6 m/s2. Airplane B is flying at the same altitude as airplane A and, as it rounds a pylon, is following a circular path of 2000-m radius. Knowing that at the given instant the speed of B is being decreased at the rate of 2 m/s2 determine, for the positions shown, (a) the velocity of B relative to A, (b) the acceleration of B relative to A.
A car travels at 100 km/h on a straight road of increasing grade whose vertical profile can be approximated by the equation shown. When the car’s horizontal coordinate is x = 400 m, what are the tangential and normal components of the car’s acceleration?
Polar and Cylindrical – Radial and Transverse
errerra
ererv
r
r
ˆ2ˆ
ˆˆ
2
1 and , ,
1
t
CtzBt
t
Ar
The three-dimensional motion of a particle is defined by the cylindrical coordinates
Determine the magnitudes of the velocity and acceleration when(a) t = 0 and (b) t = infinity
Dependent Motion
• Measure positions with respect to a fixed point.
• There will generally be a physical constraint, often a rope or cable.
At the instant shown, slider block B is moving to the right with a constant acceleration, and its speed is 6 in./s. Knowing that after slider block A has moved 10 in. to the right its velocity is 2.4 in./s, determine the accelerations of A and B.
Slider block B starts from rest and moves to the right with a constant acceleration of 1 ft/s/s. Determine the relative acceleration of portion C of the cable with respect to slider block A.
Relative Motion
• Notation – the position of B relative to A
ABr
Racing cars A and B are traveling on circular portions of a race track. At the instant shown, the speed of A is decreasing at the rate of 8 m/s2 and the speed of B is increasing at the rate of 3 m/s2 . For the positions shown, determine
a) the velocity of B relative to Ab) the acceleration of B relative to A.