Important Reminders
 Last Lecture
Kinematics - describing 1D motion
Relative velocity (yes, more vectors!)
 Pset #2 due here tomorrow at 10 am
 Today
More dimensions
Finish Mastering Physics #3 before next Monday at
10pm
More examples
Kinematics: Description of Motion
All measurements require an origin, a coordinate
system, and units
Next complication is “reference frame”, the term used to
describe the motion of observer
Constant velocity is OK, accelerated observer is not
 Basic definitions:
More vectors
 Important Concepts
 Exam #1 is next Friday at 10am
Change=derivative=slope
Position
Distance versus displacement
Multiple dimensions are as independent as many objects
Velocity - change of position
Speed is the magnitude of velocity
Think carefully about directions (changes the +/- sign)
Acceleration - change of velocity
More complicated situations
 More objects
Write an additional set of equations
 More dimensions
Write an additional set of equations
dx
v =
x dt
dy
v =
y dt
dv
d2x
a = x = 2
x
dt
dt
dv
2
y d y
a =
= 2
y
dt
dt
Vector Connections
!
! dr
v=
dt
!
!
! dv d 2 r
a=
=
dt dt 2
!
v = vx2 + vy2 + vz2
!
a = ax2 + ay2 + az2
Multi-dimensional Kinematics Problems
 Need to think carefully about directions (signs!)
 Need to think carefully about initial conditions
 Write separate equations for each dimension
 Read problem carefully to understand the specific
constraint to use to solve
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Special Case of Constant Acceleration
x = x0 + v0 x t + 12 ax t 2
vx = v0 x + ax t
y = y0 + v0 yt + 12 ayt 2
vy = v0 y + ayt
Physics
Initial conditions
Quadratic Equations
ax + bx + c = 0
2
x=
!b ± b ! 4ac
2a
2
Extra special case
Super special case
Trajectories with gravity near the surface of the Earth
and no air resistance or other drag forces
Range of a projectile near the surface of the Earth
and no air resistance or other drag forces
x0 = 0 y0 = 0 y final = 0 x final = Range
Y
v0
θ
X
ax = 0
v0 x = v0 cos(! )
ay = !g
v0 y = v0 sin(" )
Range =
v02 sin(2! )
g
You should immediately forget you ever saw this formula
but remember the technique used to find it.
Summary
 Study special cases (like range of a projectile) but
understand the assumptions that go into all formulas
 Position, velocity, and acceleration are ALL vectors
and need to be manipulated using either arrows
(qualitative) or components (quantitative)
Directions (or signs in 1D) of position, velocity, and
acceleration can all be different
Important property: Such equations can have 0, 1, or 2 solutions
depending on the value of b2 -4ac.
Negative: 0 solutions Zero: 1 solution Positive: 2 solutions
Warning: Only one of the 2 solutions may be physical!
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