Lesson 4 : Kinematics I
I.
Definitions
A. Kinematics is the ___________________ of ____________________.
B. Particle is an object of ____________________ ___________________ and _____________
__________________ ____________________ (point).
C. Position Vector
1.
Definition:
The vector that defines the ____________________ of a _________________ with respect to
a ___________________ _________________.
2.
Symbol -
3.
Units -
4.
2-D Graphical and Analytical Representation
y
Ball
x
5.
The position vector is __________ ____________ as it depends on the arbitrary choice of
_______________________ _______________________
Example: Joe and Sue determine the location pf a baseball using their own coordinate axis shown
below:
y
y'
6m
4m
4m
2m
Ball
x'
2m
Sue
4m
x
Joe
4m
8m
a) Draw the baseball's position vector as seen by Joe
b) Write the baseball's position vector seen by Joe in Cartesian form.
c)
Draw the baseball's position vector as seen by Sue
d) Write the baseball's position vector as seen by Sue in Cartesian form.
Question:
How can a baseball have two different position vectors when it only has one location?
Answer:
The position vector is a mathematical concept for describing a particle and Not A Real
(physical) quantity! What is required to correctly describe the particle is a formula relating position
vectors measured by two different observers (Joe and Sue).
Position Transformation Equation
Equations like the one above that relate the measurements of two observers are called
____________________________ _____________________ . They are very important in the field of
physics.
D. Displacement Vector
1.
Definition:
The __________________ in the __________________ __________________ of a particle.
2.
Symbol -
3.
Units -
4.
Formula:
2-D Problems
From our study of vectors, we have that for 2-D problems:
y
y2
y1

r1

r2
x
x1
x2
Example: A baseball is initially located 50 feet from the batter at a height of 5 feet. A short time
latter the ball is 30 feet from the batter at a height of 4 feet. What is the baseball's displacement?
y
x
30
50
Note: Graphically, the displacement vector is the vector that you have to add to position vector 1 to
get to position vector 2.
Displacement is NOT the same as distance! Distance is ALWAYS a positive scalar quantity while
displacement is a vector quantity.
Example:
point A.
A runner runs 100m from point A to point B. The runner then runs 100m from point B to
y
x
B
A
a) What is the distance covered?
b) What is the runner's displacement?
E. Average Velocity
1.
Definition:
The average time rate of change of the position vector.
equivalently
The displacement vector divided by the change in time.
2.
Symbol -
3.
Units -
4.
Formula -
5.
Important Facts:
a) To calculate the average velocity, you must first find the displacement vector.
b) The direction of the average velocity is the SAME as the direction of the displacement vector.
Reason: Dividing by t is the same as multiplying by the scalar (1/t) which is > 0 !!
c)
Velocity is a VECTOR and NOT the same as Speed!
Speed 
Example: Assume that our runner in the previous example covered the distance in 30s.
a) What was the runner's speed?
b) What was the runner's average velocity?
Question: What is the average velocity and speed of a runner who runs the 400m on a circular track in
50s?
F. Instantaneous Velocity
1.
Definition:
The ______________ _____________ of ______________ of the _______________________
___________________________.
Note: Unless specified otherwise in a problem, velocity means instantaneous VELOCITY.
2.
Symbol -
3.
Units -
4.
Formula -
5.
Graphical Representation: For 1-D motion, the velocity of an object at a specific point in time is the
__________________ of the __________________ ___________________ on a position-time graph
at that point.
Example:

If the position of a baseball is described by r (t)  3 t î  5t ĵ  7 k̂ ,
2
a) what is the general expression describing the velocity of the baseball as a function of time?
b) What is the velocity of the baseball at t=2s?
6.
Because velocity is defined in terms of the position vector, it depends on the observer's frame of
reference (coordinate axis).
Example: Nolan Ryan throws his 100mph fast ball while traveling on a train moving toward the
batter at 50mph. (Use your everyday intuition!)
a) What is the velocity of the fast ball according to Nolan Ryan?
b) What is the velocity of the ball according to the batter?
7.
Question:
How can we prove this?
Answer:
Using Calculus and our position transformation equation.
Velocity Transformation Equation (Derivation)
Example:
Use the velocity transformation equation to show that our previous work was correct?
Example:
ball
If Nolan Ryan is blasting off on a rocket traveling upward at 50 mph when he throws the
100 mph
a) What is the velocity of the ball as seen by the batter?
b) What is the speed of the ball as seen by the batter (neglect gravity)?
8.
Important: You can't measure absolute motion of a body! You can only measure its motion
relative to your axis! (Relative Velocity)
This is called Gallilean Relativity after Gallileo Gallile. It was the study of this motion and E&M that
lead Albert Einstein to determine the equations which produced the atomic bomb, lasers, and
computers!
Finally, it is the observer's coordinate axis that is relative and not the answer to the measurement!
Using your measurements and the transformation equations, you can exactly predict what any other
observer would see.