Motion in 1D - Physics

[Pages:16]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

1D - 6

Examples of constant acceleration in 1D

Situation I

1 2

An object starts at rest, then moves to the right (+ direction) with constant acceleration, going faster and faster.

3

4

v

2 1

4

3 v

t

a > 0, a = constant (a constant, since v vs. t is straight )

t

Situation II

An object starts at rest, then moves to the left (? direction) with constant acceleration, going faster and faster.

4

3

2

1

v 1 2

t

v 3

t

a < 0, a = constant ( since v vs. t has constant, negative slope )

4

Situation III

1

2

3

5

4

v 1 2

a < 0, a = constant !! ( since v vs. t has constant, negative slope )

t 3

4 5

9/28/2013 Dubson Notes

University of Colorado at Boulder

1D - 7

The direction of the acceleration

For 1D motion, the acceleration, like the velocity, has a sign ( + or ? ). Just as with velocity, we say that positive acceleration is acceleration to the right, and negative acceleration is acceleration to the left. But what is it, exactly, that is pointing right or left when we talk about the direction of the acceleration?

Acceleration and velocity are both examples of vector quantities. They are mathematical objects that have both a magnitude (size) and a direction. We often represent vector quantities by putting a little arrow over the symbol, like v or a .

direction of a direction of v

direction of a = the direction toward which the velocity is tending direction of v

Reconsider Situation I (previous page)

1

2

1 is an earlier time, 2 is a later time

v1 = velocity at time 1 = vinit v2 = velocity at time 2 = vfinal

v = "change vector" = how v1 must be "stretched" to change it into v2

v1 v v2

direction of a = direction of v

Situation II: Situation III:

v v1 v2 v1

In both situations II and III, v is to the left, so acceleration a is to the left

v2

v

( This has been a preview of Chapter 3, a d v ) dt

Our mantra: " Acceleration is not velocity, velocity is not acceleration."

9/28/2013 Dubson Notes

University of Colorado at Boulder

1D - 8

Constant acceleration formulas (1D)

In the special case of constant acceleration (a = constant), there are a set of formulas that relate position x, velocity v, and time t to acceleration a.

formula

relates

(a) v vo a t

(v, t)

(b) x xo vo t (1/ 2) a t2

(c)

v2

vo2 2a (x xo )

(x, t) (v, x)

(d) v

vo v 2

xo , vo = initial position, initial velocity x, v = position, velocity at time t

Reminder: all of these formulas are only valid if a = constant, so these are special case formulas. They are not laws. (Laws are always true.)

dv

Proof of formula (a) v vo a t . Start with definition a

. dt

In the case of constant acceleration, a a

v

v2 v1

t

t2 t1

Since a = constant, there is no difference between average acceleration a and instantaneous

acceleration at any time.

v1 vo , v2 v t1 0 , t2 t

a v v0 t

v vo a t

(See the appendix or your text for proofs of the remaining formulas.)

Example: Braking car. A car is moving to the right with initial velocity vo = + 21 m/s. The brakes are applied and the car slows to a stop in t = 3 s with constant acceleration. What is the acceleration of the car during braking?

a = ?

aa

v

v v0

0 21 m / s

t

t

3s

(Do you understand why we have set v = 0 in this problem? )

7m / s2

Negative acceleration means that the acceleration is to the left.

9/28/2013 Dubson Notes

University of Colorado at Boulder

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