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

,

time elapsed

s=

d

,

t

units are m/s or mph or km/hr or...

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 +/¨C sign

+ = going right ?

always!

¨C = going left ?

vA = ¨C10 m/s

A

vB = +10 m/s

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 ?

x1

change in position

change in time

?

?x

?t

x2

x

0 (initial)

v ?

xf ? xi

tf ? ti

?

(final)

x 2 ? x1

t 2 ? t1

?

?x

?t

?x = xfinal ¨C xinitial = displacement (can be + or ¨C )

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? University of Colorado at Boulder

1D - 2

Notice that ? (delta) always means "final minus initial".

?x

is the slope of a graph of x vs. t

?t

v ?

Review: Slope of a line

y

(x2, y2)

rise

slope = run

?y

(x1, y1)

=

?y

?x

y2 ¨C y1

x2 ¨C x1

=

?x

x

y

y

y

0 slope

(¨C) slope

(+) slope

x

x

x

Suppose we travel along the x-axis, in the positive direction, at constant velocity v:

start

x

0

x

rise

slope = run

x2

=

?y

?x

=

?x

= v

?t

?x

y-axis is x, x-axis is t .

x1

?t

t1

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t2

t

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1D - 3

Now, let us travel in the negative direction, to the left, at constant velocity.

start

x

0

x

?x

slope = v =

0 (fast)

t

The slope at a point on the x vs. t curve is the instantaneous velocity at that point.

x

?t

?x

?x

?t

t

Definition: instantaneous velocity = velocity averaged over a very, very short (infinitesimal)

time interval

?x

dx

v ? lim

?

= 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

dx

?x

? lim

defined as the slope of the tangent line:

.

?t ? 0 ?t

dt

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1D - 4

x

tangent line

?t

?x

t

x

t

slow

fast

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

?t ?0

?v

dv

?

?t

dt

average acceleration over a non-infinitesimal time interval ?t : a ?

units of a = [a] ?

m/s

s

?

?v

?t

m

s2

?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.

Sometimes I will be a bit sloppy and just write a ?

9/28/2013 Dubson Notes

? University of Colorado at Boulder

1D - 5

a ?

dv

dt

?v

?

?t

v f ? vi

tf ? ti

?

v2 ? v1

t 2 ? 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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