Motion in 1D - Physics
1D - 1
Motion in one dimension (1D)
In this chapter, we study speed, velocity, and acceleration for motion in one-dimension. One dimensional motion is motion along a straight line, like the motion of a glider on an airtrack.
speed and velocity
speed distance traveled , s = d , units are m/s or mph or km/hr or...
time elapsed
t
speed s and distance d are both always positive quantities, by definition.
velocity = speed + direction of motion Things that have both a magnitude and a direction are called vectors. More on vectors in Ch.3.
For 1D motion (motion along a straight line, like on an air track), we can represent the direction of motion with a +/? sign
+ = going right ? = going left
always!
vA = ?10 m/s
vB = +10 m/s
A
B
x
0
Objects A and B have the same speed s = |v| = +10 m/s, but they have different velocities.
If the velocity of an object varies over time, then we must distinguish between the average velocity during a time interval and the instantaneous velocity at a particular time.
Definition: average velocity = v change in position x
change in time
t
x1
x2
x
0 (initial)
(final)
v xf xi x2 x1 x
tf ti
t2 t1
t
x = xfinal ? xinitial = displacement (can be + or ? )
9/28/2013 Dubson Notes
University of Colorado at Boulder
Notice that (delta) always means "final minus initial".
v x is the slope of a graph of x vs. t t
Review: Slope of a line
y (x2, y2)
(x1, y1)
y
slope =
rise run
=
y x
=
y2 ? y1 x2 ? x1
x x
1D - 2
y (+) slope x
y (?) slope x
y 0 slope x
Suppose we travel along the x-axis, in the positive direction, at constant velocity v:
start
x 0
x x2
slope =
rise run
=
y x
=
x = v
t
x
x1 t
y-axis is x, x-axis is t .
t1
t2
t
9/28/2013 Dubson Notes
University of Colorado at Boulder
Now, let us travel in the negative direction, to the left, at constant velocity.
start
x 0
x
x
slope = v =
< 0
t
t
t
x < 0
1D - 3
Note that v = constant slope of x vs. t = constant graph of x vs. t is a straight line
But what if v constant? If an object starts out going fast, but then slows down and stops... x
slower
slope = 0 (stopped)
slope > 0 (fast) t
The slope at a point on the x vs. t curve is the instantaneous velocity at that point. x
x t
x t
t
Definition: instantaneous velocity = velocity averaged over a very, very short (infinitesimal) time interval
v lim x d x = slope of tangent line. In Calculus class, we would say that the
t 0 t
dt
velocity is the derivative of the position with respect to time. The derivative of a function x(t) is
defined as the slope of the tangent line: d x lim x .
dt
t 0 t
9/28/2013 Dubson Notes
University of Colorado at Boulder
x
tangent line
x t
x
1D - 4 t
t
fast
slow
v
= dx/dt
t
Acceleration
If the velocity is changing, then there is non-zero acceleration.
Definition: acceleration = time rate of change of velocity = derivative of velocity with respect to time
In 1D: instantaneous acceleration a lim v d v
t0 t
dt
average acceleration over a non-infinitesimal time interval t : a v t
units of a = [a]
m/s s
m s2
Sometimes I will be a bit sloppy and just write a v , where it understood that t is either a t
infinitesimal time interval in the case of instantaneous a or t is a large time interval in the case
of average a.
9/28/2013 Dubson Notes
University of Colorado at Boulder
1D - 5
a dv dt
v vf vi v2 v1
t
tf ti
t2 t1
v = constant v = 0 a = 0
v increasing (becoming more positive) a > 0
v decreasing (becoming more negative) a < 0
In 1D, acceleration a is the slope of the graph of v vs. t (just like v = slope of x vs. t )
Examples of constant acceleration in 1D on next page...
9/28/2013 Dubson Notes
University of Colorado at Boulder
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