Chapter 2 – Motion in one Dimension
2-1 Displacement and Velocity
page 40
Getting from here to there always requires motion. The simplest form of motion is onedimensional motion, motion that occurs in only one direction.
Motion – the movement of an object from one position to another
1. Takes place over time.
2. There is a change in position
Frame of reference – a coordinate system for specifying the precise location of objects in space
1. A point in space from which displacement can be observed or measured.
Examples: mile markers or highway mileage signs – a car passes these frames of reference
2. An object that is at rest does not have a change in position in reference to a frame of
reference.
Displacement – the straight line length that an object moves from its initial position to its final
position
What are the SI units for displacement (length)?
It does not matter what circuitous route an object travels to get from point “A” to point “B.” The
displacement will still be the straight-line distance between point “A” and point “B.”
Displacement is therefore not always equal to the distance traveled. Displacement may be either
a positive or negative number. The change in the sign (+ or -) can also indicate direction or
perhaps a change in direction. If the X-axis is orientated such that positive numbers go towards
North and negative number go towards South, direction of displacement is indicated.
The formula that may be used to determine the amount of displacement along the X-axis is:
ΔX = Xf – Xi
ΔX – change in position (displacement)
Xf – final position
Xi – initial position
If the direction changes to the Y-axis, then “Y” is substituted for the “X.”
If you begin at the trailhead on a hike through the mountains, it does not matter your direction,
East, West, up, or down, if you return to the starting point, your displacement is zero.
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Average velocity – total displacement divided by the time interval
Any time interval can be selected for calculation of the average velocity. To determine the time
interval:
Δt = tf – ti
Δt – change in time
tf – final time
ti – initial time
Note: Speed is a scalar quantity and velocity is a vector quantity. What does that mean? It
means that speed is a numerical quantity without direction while velocity is a numerical quantity
that includes direction. Both are sometimes interchanged when the only purpose is to describe
magnitude of the moving object.
Scalar quantity – a numerical quantity (speed, mass, time) that does not indicate a direction
Vector quantity – a numerical quantity that has a direction (N, SE, 45 degrees, up, 180)
The units, time (t) are measured in seconds (s) and distance (d) is measured in meters (m). The
d
fundamental formula for speed is: s =
and the SI units for speed is meters/second (m/s).
t
x f  xi
x
Study the equation for average velocity in one dimension. Vavg =
=
t
t f  ti
What are the SI units for velocity?
Velocity on a Graph – Position is plotted on the vertical axis and time is plotted on the
horizontal axis. The average velocity will be the slope of the line on the graph as represented by
rise
the following equation when the coordinates are taken from the graph.
slope =
run
Rise is the change in the vertical coordinates and run is the change in the horizontal
coordinates. One graph can also be used to plot the velocity of multiple objects.
Instantaneous Velocity – the velocity of an object at some instant (or specific point in its path)
The slope of a line tangent to the curve of the slope at point t will be the instantaneous velocity.
An example of instantaneous velocity would be the number indicated on the speedometer of a
car at the instant you look at it.
Why would we need to know the instantaneous velocity of an object? We know that an object
can never reach instantaneous velocity without accelerating or decelerating. We may need
to observe or define the behavior of an object at a given moment in time.
Instantaneous Speed - the magnitude of instantaneous velocity
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2-2 Acceleration
page 48
Very uncommon it is that an object can always travel at an exact constant velocity. They will be
moments of slowing down or speeding up. The sensation of slowing or speeding up in a car can
be felt as a driver pushes on the brake or pushes on the gas.
Acceleration – the rate of change of velocity for a given time interval
1. The magnitude of acceleration is calculated by the change in velocity by the time interval
2. Acceleration is a vector quantity because it has direction.
3. If there is no change in velocity ( Δv = 0 m/s ), then there is no acceleration.
The formula for average acceleration is:
aavg =
v f  vi
v
=
t f  ti
t
aavg – average acceleration
Δv – change in velocity
Δt – change in time
Show how the SI units for acceleration are derived.
m/s
m 1 m
= x  2
s
s s s
Note: We can never go back in time. Well, Peabody can in his “Wayback” machine. All values
of time will be positive. Remember this fact when working through any problems.
Graphing Acceleration
Velocity is always on the Y-axis and time is always on the X-axis. It is permissible to have
negative and positive values acceleration.
Motion with Constant Acceleration – The relationship between displacement, velocity, and
constant acceleration are expressed by equations that apply to any object moving with constant
acceleration. The equations listed in the table on page 58 can solve any one-dimensional motion
problem that has uniform acceleration.
Note: Do not try to memorize all these equations. Do learn the fundamental equations for
velocity and acceleration. For any unknown in distance, time, velocity, and acceleration, derive
the formula you need by algebraic rearrangement or formula substitution.
Velocity formula: vavg =
x
t
acceleration formula: aavg =
v
t
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2-3 Falling Objects
page 60
Free Fall – the motion of an object falling with a constant acceleration
1. Objects at near surface earth will accelerate towards Earth due to the pull of gravity at 9.81
m/s2.
2. Acceleration, a = -g = 9.81m/s2 on Earth; in any equation “g” is a constant – the acceleration
of an object due to gravity
3. The factor that does not allow a feather to fall at the same rate as a ball is air resistance. If the
objects are falling in a vacuum, then they will fall at the same rate.
Since all objects (most all objects) fall to Earth at a constant acceleration of “g”, the velocity
proportionally increases to the time. The following formula will determine the instantaneous
velocity at a point in time.
v = gt
v = velocity in m/s
g = gravity (9.81 m/s2)
t = time in seconds
What is the velocity of the falling object after one second?
What is the velocity of the falling object after two seconds?
Why is this velocity and not speed?
At any point in time the distance an object has fallen toward Earth can be calculated. By
algebraically rearranging the above formula to solve for distance you have:
d = ½ gt2
d = distance object has fallen (meters)
g = gravity (meters per second squared)
t = time (seconds)
What is the distance the object has fallen after one second?
What is the distance the object has fallen after two seconds?
Is this a scalar or vector quantity? Why?
Throw a ball vertically into the air. At the top of the trajectory the velocity will be ______. Why?
What is the acceleration due to gravity at the top of the trajectory? (Think: Does the force of gravity stop
acting on the ball at the top of the trajectory? Hmmmmmmm!)
Terminal velocity – the maximum velocity an object can achieve as it falls towards Earth
As an object falls towards Earth, at some point, the air will push up with a force that is equal to
the pull of gravity. At that point acceleration towards Earth will stop and the object will continue
to fall to Earth at a constant velocity, the terminal velocity.
“g” will also be different on different planets. On the moon the pull of gravity is approximately
one-sixth that of the Earth.
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Momentum
page 208
What is the velocity of that moving mass? Is the momentum of the object related more to the
mass or the velocity?
Momentum – a vector quantity defined as the product of an object’s mass and velocity
The formula for linear momentum is: p = mv
The SI units for momentum is kgm/s
A tiny object that falls from a very high altitude can cause considerable damage to Earth
structures … or a person’s head! Ouch!
A few hints for solving physics problems:
1. Make an abbreviated sketch of the event. This helps to organize the event know, the givens,
and the unknowns.
2. List the known quantities.
Example:
ti = 5 s
tf = 10 s
x = 25 m
3. List the unknown quantities (or the one(s) you asked to solve for).
Example:
vavg = _____
4. Use dimensional analysis to convert units as needed.
Example:
km to meters, hours to minutes or seconds, kilograms to grams, etc
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