Phy 2053 Chapter sections 2.5-2.6 and 3.1-3.4
Kinematics, 1-D constant acceleration motion,
Free Fall, Vectors, and Motion in 2-D
Thursday, Sept. 8, 2007—Gary Ihas
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1.
2.
Some examples of Use
 v  vf 
Zero acceleration: x  v average t   o
t
 2 
Displacement not needed:
Kinematic Equations
Describe Motion of an Object-ball, rocket, person…
Used in situations with uniform acceleration a
Equation #1 (velocity) v  vo  αt

v0 + v
2
Vel. at
time t
x v0 + v
v=
=
t
2
v +v
1
 v + v + at 
x= 0
t  v0t  at 2
t =  0 0

2


2
2


v = vaverage =
Kinematic Equations—1-2
Constant Acceleration
Eq. 1 v  vo  at
Eq. 2
v  vo  at
Eq. 3
Final velocity not needed:
or t =
- and 3
v - v0
a
1 2 Putting t from Eq. 1
at
2
into Eq. 2 we get
v 2  vo2  2ax Eq.3
x  vot 
Note Eq. 3 is not independent of Eq. 1 and Eq. 2
½at2
v0t
1
x  v o t  at 2
2
Time not needed: v 2  v o2  2ax
Free Fall



If only force on object moving near surface of
earth is gravity it is in free fall
Free fall is constant acceleration
The acceleration is called the acceleration due
to gravity, symbolized by g

g = 9.80 m/s²

g is always directed downward




When estimating, use g  10 m/s2
Free Fall options
• Initial velocity is zero
v=0
• Throw up -initial velocity
non-zero and positive—
instantaneous velocity at
maximum height = 0
• Throw down –initial
velocity is negative
toward the center of the earth
Ignore air resistance and assume g doesn’t
vary with altitude over short vertical distances
Forces can apply to the object before or after the free fall
• Starting and ending heights may
be equal – symmetric or
not equal-asymmetric - trajectory
1
If we lived in 1-Dimension, we would not need vectors, but…
Would you like to be always standing in line, and the person in
front of you only changed if he (or she) died?
Problem-Solving Hints


Read the problem
Draw a diagram


Label all quantities, be sure all the units are
consistent




Convert if necessary
Choose the appropriate kinematic equation
Solve for the unknowns

2-D Rules!
Choose a coordinate system, label initial and final
points, indicate a positive direction for velocities and
accelerations
Vectors let us keep track of our way in any number of dimensions
by carrying all the numbers-components-we need in one symbol


You may have to solve two equations for two
unknowns
Check your results


and y-axes
Estimate and compare
Check units
Ax
The x-component of a vector
is the projection along the
 x-axis

A x  A cos  x
The y-component of a vector is the
projection along the y-axis


 

Then, A  Ax  Ay
A y  A sin  y
Ay are scalars
Note that vectors are
unchanged by being moved
as long as their direction or
magnitude is not changed
These equations are valid only if θ is
measured with respect to the x-axis
Special case of vector
addition--add the
negative of the
subtracted vector
 


A  B  A  B
 

Ay
When you add vectors, just put
the tail of one on the head on
the next…

 The resultant R is drawn from
the origin of the first vector to
the end of the last
vectorVectors obey the
commutative law of addition
The order in which the vectors
are added
 doesn’t

affect
 the
result A  B  B  A
Vector Subtraction and Scalar Multiplication

and


Ay  A sin 
and
Ax
Graphically Adding Vectors
Vector
Components
Ax  A cos 
Here is a vector that shows
us going at an angle  with
the x axis
It is useful to use
rectangular components
to manipulate vectors
 These are the projections
of the vector along the x-
Continue with standard
vector addition procedure
The result of the multiplication or division of a vector by a scalar is a
vector-the magnitude of the vector is multiplied or divided by the scalar
•If the scalar is positive, the direction of the result is the same as of the
original vector
•If the scalar is negative, the direction of the result is opposite that of the
original vector
Adding Vectors Algebraically


Choose a coordinate system and sketch the
vectors
Find the x- and y-components of all the vectors
Add all the x-components
Add all the y-components
R x   Ax
R y   Ay
Use Pythagorean theorem find
The magnitude of the resultant:
R= R 2x +R 2y
Use inverse tangent function
To find the direction of R:
  tan 1
Ry
Rx
2