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Chapter 2 – Describing Motion: Kinematics in One Dimension
September 1, 2011
Mechanics: study of the motion of objects, and the related concepts of force and energy,
can be divided into two parts
Kinematics: description of how objects move
Dynamics: deals with force and why objects move as they do
Translation motion: objects that move without rotating
Point particle: mathematical point and to have no spatial extent (no size)
can only undergo translational motion
Particle model is useful when the object’s size is not so significant
Basic Concepts in Kinematics:
Distance [m] : SCALAR
Displacement [m] : VECTORS
Speed [m/s] : SCALAR
Velocity [m/s] : VECTORS
Reference Frame and Displacement:
Any measurement of position, distance or speed must be made with respect to a reference
frame or frame of reference.
Example: A person walks toward the front of a train at 5 km/h. The train is moving
80 km/h with respect to the ground, so the walking person’s speed relative to the
ground is 85km/h.
In physics, we often draw a set of coordinate axes to specify the direction of the motion
and to represent a frame of reference. In three dimensions a z-axis perpendicular to the x
and y axes is added.
For one-dimensional motion, we often use the x axis as the line along which the motion
takes place. Then the position of an object at any moment is given by its x coordinate. If the
motion is vertical, as for a dropped object, we usually use the y-axis.
Distance: total length of the trajectory Displacement: the net change in position of the object, how far the object is from its
starting point
Example: A person walks 70m east, the 30m west. The total distance traveled is
100m but the displacement, shown as a blue arrow, is 40m to the east.
Displacement is a quantity that has both magnitude and direction, such quantities are
called vectors.
x1 x2 Δx = x2– x1
10 m 30 m
Δ means final value minus initial value (change in)
Average Velocity:
The term speed refers to how far an object travels in a given time interval, regardless of
direction. Average speed is defined by the total distance traveled along its path by the time
it takes to travel this distance.
Average Speed = Distance Traveled/Time Elapsed
Speed is a positive number, with units. Velocity is used to signify both the magnitude of
how fast an object is moving and also the direction in which it is moving (therefore a
vector). Average velocity is defined in terms of displacement, rather than total distance
traveled.
Average Velocity = Displacement/Time Elapsed
= Final Position – Initial Position/Time Elapsed
The elapsed time, or time interval, t2– t1is the time that has passed during our chosen
period of observation 2.3 - Instantaneous Velocity:
The velocity at any given time, the average velocity during an infinitesimally short time
interval.
v = lim Δx
Δt 0 Δt
Can displacement and velocity have different signs (ie, be in different directions)?
Yes
2.4 – Acceleration
Acceleration specifies how rapidly the velocity of an object is changing. Average
Acceleration is defined as the change in velocity divided by the time taken to make this
change:
Average Acceleration = Change of velocity/Time Elapsed
Acceleration is also a vector, but for one-dimensional motion, we need only use a plus or
minus sign to indicate direction relation to a chosen coordinate system.
Instantaneous Acceleration can be defined in analogy to instantaneous velocity:
a = lim Δv
Δt 0 Δt
Normal units of acceleration in MKS:
m/s 2
acceleration due to gravity 9.8 m/s
Decleration is often used when an object is slowing down, means that the magnitude of the
velocity is decreasing; it does not necessarily mean it is negative
2.7 – Falling Objects Galileo Galilei (father of modern science) discovered:
o The speed of a falling object is NOT proportional to its mass or weight
o All objects, light or heavy, fall with the same acceleration, in the absence of air
Example: A ball and a light piece of paper are dropped at the same
time – The baseball hits the ground first. Repeated, with the paper
wadded up – They both reach the ground at the same time.
Galileo’s hypothesis: free fall is at constant acceleration – at a given location on the Earth
and in the absence of air resistance, all objects fall with the same constant acceleration. We
call this acceleration due to gravity on the Earth (represented by the symbol g) with an
approx. magnitude of 9.80 m/s 2
2.8 – Graphical Analysis of Linear Motion
Example: Graph the displacement: time is independent variable (x-axis) and displacement
is on the y-axis
Slope = (change in displacement)/(change in time) = velocity!
Displacement-vs-time graphs can be used to calculate the velocity-vs-time graphs
Quiz Results
Reshmi & Sarah on their bikes:
Velocity vs. Time graph
If displacement’s are the same, R is moving faster
Sarah will always win short distance races
Reshmi will always win long distance races
Therefore you cannot tell who will win
Velocity
Time 34, 501, 798 / 80 seconds
Best answer: 34.5 Million people
Quiz #2:
Which of t

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