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**Unformatted text preview: **t that it interacts with other objects
The stronger the interaction, the faster the change in velocity
• (Object at rest is a case of uniform velocity) Does this match what you see
in the REAL WORLD?? Vectors have Magnitude and Direction
In three dimensions and Cartesian coordinates: Vectors
!
ˆ
ˆ
r = x, y, z = rx , ry , rz = rx i + ry ˆ + rz k
j
ˆ
i = 1, 0, 0
ˆ = 0,1, 0
j
ˆ
k = 0, 0,1 Unit vectors in
the direction of
the axes: General unit vector: !
r
ˆ
r= ! =
r x, y , z (x 2 +y +z
2 2 ) HOME READING: OPERATIONS WITH VECTORS = ||r
rˆ coolest trick
in the book! Vectors vector magnitudes
scalar multiplication Velocity has Magnitude and Direction
Magnitude of Velocity = Speed (a scalar) 100 m in 10 s
Average speed: vavg = d 100 m
=
= 10 m/s
t
10 s If we know speed we can predict future: d = vavg t = 10 m/s ⋅ 10 s = 100 m
If we know speed we can reconstruct past: t = d
100 m
=
= 10 s
vavg 10 m/s Velocity has Magnitude and Direction
Velocity is a Vector
!""
!r = rf " ri !
v z !y
ri x 100 m in 10 s Definition:
average velocity vavg d
=
t !
vavg !
rf !!
! r #r
Δrx Δry Δrz
!r
f
i
≡
,
,
=
"
!t t f # ti
Δt Δt Δt Example y
9
8m
7 !
ri
-2 -1 6
5
4
3
2 !
!r !
rf
!
vavg x 1 2 3 4 5 6 7m !!
! r #r
!r
=
"f i
!t t f # ti Instantaneous vs. average velocity
The trajectory of a ball through air:
!
vB Instantaneous
velocity at point B It is tangent to
trajectory at point B !
vCB
!
v DB !
vEB It's the SLOPE! The average velocity will depend on the choice of !
!r Instantaneous velocity: !
!
!
!r d r
v = lim
#
!t "0 !t
dt derivative and Δt Acceleration = Change in Velocity
d
v
=
a
dt
Express
v as = | |v
v
vˆ Now use the chain rule to take the derivative: =
a d
v
d
=
dt
dt =
| |v
vˆ
d| |
v
v+
ˆ
dt Rate of change of direction
d| v |
ˆ
| |
v
dt Rate of change of magnitude of velocity (speed)
is parallel to the velocity. Predicting new position !
vavg !!
! r #r
!r
=
"f i
!t t f # ti ( !!!
rf ! ri = vavg t f ! ti ) The position update formula ! !!
rf = ri + vavg (t f ! ti ) Interactions: changing velocity
Newton s first law of motion is qualitative:
An object moves in a straight line and at constant speed
except to the extent that it interacts with other objects Interactions can change velocity!
? What factors make it difficult to change an object velocity? Mass! Introduce new parameter that involves product of mass and velocity: momentum
The stronger the interaction, the larger the change in momentum !
"
"
The simplest way: p = mv
MOMENTUM (Legal Disclaimer: there's more to momentum
for objects near the speed of light! Void in NH.) Average rate of change of momentum The stronger the interaction, the faster is the change in the momentum Average rate of change of momentum: !
!
! p "p
!p
f
i
=
!t
t f " ti Instantaneous rate of change of momentum: !
!
dp
!p
= lim
dt !t"0 !t Units: kg ⋅ m
s2 The principle of relativity
Physical laws work in the same way for observers
in uniform motion as for observer at rest Inertial reference frame Inertial frame moves at constant velocity. Physical laws work in the same way in any inertial frame Are you in an inertial reference frame right now? Special theory of relativity
Inertial frame moves at constant velocity. Speed of light = constant in all inertial reference frames! SPACE AND TIME WARP
TO ENSURE THIS STAYS TRUE Time dilation: time runs slower in moving reference frames
Length contraction: object length becomes shorter in moving reference frame Momentum – The Whole Story
Definition of momentum: !
!
p = ! mv != 1
!2
"v%
1! $ '
# c& (Lorentz factor) c = speed of light ≈ 3 ×108 m / s !
"
"
For v << c, γ≈1, approximation: p = mv
γ v, m/s
0
300 p= 1
1.0000000000005 30,000 1.000000005 3×107 1.005 0.9999c 70.7
No mass can reach speed of light! YOU ARE HERE
p= What We Did Today • General Course Information
• Newton's First Law of Motion
• Vectors
• Velocity
• Principle of Relativity
• Special Relativity...

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