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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