Physics
Constant Acceleration Particle Models
Unit 3
Unit Essential Questions
What are the descriptive models of motion and how are they used?
What are the graphical models and the mathematical models that are used to represent the
motion of an object having constant acceleration and how are these models used to
analyze the object’s motion?
What is the relationship between a ball’s position and the clock reading as the ball rolls
down an incline?
What is the relationship between a ball’s velocity and the clock reading as the ball rolls
down an incline?
What is the relationship between a ball’s velocity and the ball’s position as the ball rolls
down an incline?
What is the motion of a fan powered cart?
What is the motion of a freefalling object?
New Understandings
A. A constantly accelerating object’s position is directly proportional to the clock
reading squared.
B. A constantly accelerating object’s velocity is directly proportional to the clock
reading.
C. Acceleration is defined as the slope of a velocity vs time graph.
D. A constantly accelerating object’s velocity is directly proportional to the square root
of the object’s position.
E. General mathematical models can be derived from specific mathematical models.
F. Symbols are used in math models to represent both constants and variables; constants
are treated the same as variables in algebraic manipulations, but constants are treated
differently than variables in calculus manipulations.
G. To properly apply a general mathematical model for the analysis of a physical system,
the user must be able to distinguish the constants from the variables in the
mathematical model.
H. A small number of useful general mathematical kinematic models can be derived by
analyzing and judiciously combining specific graphical and mathematical models
extracted from experimental data.
I. An object’s velocity and acceleration can be represented with vectors on a motion
map.
J. An object’s motion can be analyzed and predicted using general graphical models and
general mathematical models. Analysis with mathematical models uses algebra
concepts; analysis with graphical models uses algebra concepts and calculus
concepts.
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Physics
Constant Acceleration Particle Models
Unit 3
K. Freefall is a constant acceleration case.
L. All freefalling objects have the same acceleration, regardless of object mass.
M. An object moving away from the earth can be freefalling.
N. Graphical models and mathematical models are used to represent, analyze, and
communicate structure and relationships in physical systems and physical
interactions.
O. A relatively small number of models can be used with great versatility for a wide
variety of physical systems and physical interactions.
New Math Concepts
Geometric algebra derivation of kinematic equations, quantitative graphical solutions
using calculus concepts, vector representations, tangent, slope of tangent, limit, area
under curve, instantaneous value from average value, mean speed theorem
New Technology Skills
Using a motion detector, using photogates, using a picket fence with a photogate
Learning Targets
1. You should be able to determine the instantaneous velocity of an object in three ways:
a. determining the slope of the tangent to its position vs time graph at a given clock
reading.
b. using the mathematical model v = vo + at
c. using the mathematical model v2 = vo2 + 2ax
2. You should be able to determine the displacement of an object in three ways:
a. finding the area under its velocity vs time graph
b. using the mathematical model x = vot + 1/2 at2 + xo
c. using the mathematical model v2 = vo2 + 2ax
3. You should be able to determine the acceleration of an object in five ways:
a. determining the slope of its velocity vs time graph
b. using the mathematical model a ≡ ∆v/∆t
c. using the mathematical model x = 1/2 at2 + vot + xo
d. using the mathematical model v = vo + at
e. using the mathematical model v2 = vo2 + 2ax
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Physics
Constant Acceleration Particle Models
Unit 3
4. Given an object’s position vs time graph, you should be able to:
a. describe the motion of the object
b. draw the object’s velocity vs. time graph
c. draw the object’s acceleration v. time graph
v
v
d. draw the object’s motion map (including v and a vectors)
e. write a mathematical model that describes the motion
f. determine the object’s acceleration
g. determine the object’s displacement for a specified time interval
5. You should be able to use photogates, a computer interface, and data acquisition
software to acquire position and clock reading data for a moving object.
6. You should be able to use all of these and your prior knowledge, skills, and
understandings to represent, analyze, and communicate structure and relationships in
physical systems and physical interactions
New Terms and Symbols
v
a
acceleration
exemplar
gravitational acceleration
motion detector
photogate
uniform velocity
v
v
air resistance
freefall
kinematics
negative acceleration
picket fence
v
a
constant acceleration
g
mean speed theorem
negative displacement
uniform acceleration
New Math Models
a≡
∆v
∆t
x = 12 at 2 + v0 t + x0
v = at + v0
v 2 = 2ax + v02
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