Physics 215 -Fall 2014Lecture01-21 Welcome Back to Physics 215!
(General Physics I Honors & Majors)
Slide 2
Lecture 01-2 2 Current homework assignment HW1: Ch.1 (Knight):
42, 52, 56; Ch.2 (Knight): 26, 30, 40 due Wednesday Sept 3 rd in
recitation TA will grade each HW set with a score from 0 to 5
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Lecture 01-2 3 Workshops Two sections! Wednesdays: 8:25-9:20AM
and 2:35-3:30PM Fridays: 8:25-9:20AM and 10:35-11:30AM Both meet in
room 208 Attend either section each day Dont need to change
registration
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Velocity Definition: Average velocity in some time interval t
is given by v av = (x 2 - x 1 )/(t 2 - t 1 ) = x/ t Displacement x
can be positive or negative so can velocity it is a vector, too
Average speed is not a vector, just (distance traveled)/ t
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Discussion Average velocity is that quantity which when
multiplied by a time interval yields the net displacement For
example, driving from Syracuse Ithaca
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Instantaneous velocity But there is another type of velocity
which is useful instantaneous velocity Measures how fast my
position (displacement) is changing at some instant of time Example
-- nothing more than the reading on my cars speedometer and my
direction
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Describing motion Average velocity (for a time interval): v
average = Instantaneous velocity (at an instant in time) v instant
= v = Instantaneous speed |v|
Slide 8
Instantaneous velocity But there is another type of velocity
which is useful instantaneous velocity Measures how fast my
position (displacement) is changing at some instant of time Example
-- nothing more than the reading on my cars speedometer and my
direction
Slide 9
Describing motion Average velocity (for a time interval): v
average = Instantaneous velocity (at an instant in time) v instant
= v = Instantaneous speed |v|
Slide 10
Instantaneous velocity Velocity at a single instant of time
Tells how fast the position (vector) is changing at some instant in
time Note while x and t approach zero, their ratio is finite!
Subject of calculus was invented precisely to describe this limit
derivative of x with respect to t
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Lecture 01-2 11 Velocity from graph P Q xx tt V av = x/ t As t
gets small, Q approaches P and v dx/dt = slope of tangent at P t x
instantaneous velocity
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Lecture 01-2 12 Cart Demo Transmitter sends out signal which is
reflected back by cart - can calculate distance to cart at any
instant -- x(t) Cart is not subject to any forces on track - expect
constant velocity Computer shows position vs. time plot for motion
(and velocity plot)
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Lecture 01-2 13 When does v av = v inst ? When x(t) curve is a
straight line Tangent to curve is same at all points in time We say
that such a motion is a constant velocity motion well see that this
occurs when no forces act x t
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Lecture 01-2 14 Interpretation Slope of x(t) curve reveals v
inst (= v) Steep slope = large velocity Upwards slope from left to
right = positive velocity Average velocity = instantaneous velocity
only for motions where velocity is constant t x
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Lecture 01-2 15 Positions: x initial, x final Displacements: x
= x final - x initial Instants of time:t initial, t final Time
intervals: t = t final - t initial Average velocity:v av = x/ t
Instantaneous velocity:v = dx/dt Instantaneous speed:|v| = |dx/dt|
Summary of terms
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Lecture 01-2 16 Acceleration Similarly when velocity changes it
is useful (crucial!) to introduce acceleration a a av = v/ t = (v F
- v I )/ t Average acceleration -- keep time interval t non-zero
Instantaneous acceleration a inst = lim t 0 v/ t = dv/dt
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Lecture 01-2 17 Sample problem A cars velocity as a function of
time is given by v(t) = (3.00 m/s) + (0.100 m/s 3 ) t 2. Calculate
the avg. accel. for the time interval t = 0 to t = 5.00 s.
Calculate the instantaneous acceleration for i) t = 0; ii) t = 5.00
s.
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Lecture 01-2 18 Acceleration from graph of v(t) t v P Q T What
is a av for PQ ? QR ? RT ? R Slope measures acceleration Positive a
means v is increasing Negative a means v decreasing
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Lecture 01-2 19 Fan cart demo Attach fan to cart - provides a
constant force (well see later that this implies constant
acceleration) Depending on orientation, force acts to speed up or
slow down initial motion Sketch graphs of position, velocity, and
acceleration for cart that speeds up
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Interpreting x(t) and v(t) graphs Slope at any instant in x(t)
graph gives instantaneous velocity Slope at any instant in v(t)
graph gives instantaneous acceleration What else can we learn from
an x(t) graph? x t v t
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Lecture 01-2 21 You are throwing a ball up in the air. At its
highest point, the balls 1.Velocity v and acceleration a are zero
2.v is non-zero but a is zero 3.Acceleration is non-zero but v is
zero 4.v and a are both non-zero
Slide 22
Lecture 01-2 22 Cart on incline demo Raise one end of track so
that gravity provides constant acceleration down incline (well
study this in much more detail soon) Give cart initial velocity
directed up the incline Sketch graphs of position, velocity, and
acceleration for cart
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Lecture 01-2 23 Acceleration from x(t) plot ? If x(t) plot is
linear zero acceleration Is x(t) is curved acceleration is non-zero
If slope is decreasing a is negative If slope is increasing a is
positive If slope is constant a = 0 Acceleration is rate of change
of slope!
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Lecture 01-2 24 t x b a The graph shows 2 trains running on
parallel tracks. Which is true: 1.At time T both trains have same v
2.Both trains speed up all time 3.Both trains have same v for some
t
Lecture 01-2 25 Acceleration from x(t) Rate of change of slope
in x(t) plot equivalent to curvature of x(t) plot Mathematically,
we can write this as a = Negative curvature a < 0 Positive
curvature a > 0 No curvature a = 0 x t
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Lecture 01-2 26 Sample problem An objects position as a
function of time is given by x(t) = (3.00 m) - (2.00 m/s) t + (3.00
m/s 2 ) t 2. Calculate the avg. accel. between t = 2.00s and t =
3.00 s. Calculate the instantaneous accel. at i) t = 2.00 s; ii) t
= 3.00 s.
Slide 27
Displacement from velocity curve? Suppose we know v(t) (say as
graph), can we learn anything about x(t) ? Consider a small time
interval t v = x/ t x = v t So, total displacement is the sum of
all these small displacements x x = x = lim t 0 v(t) t = v t
Slide 28
Graphical interpretation v t TT T2T2 tt v(t) Displacement
between T 1 and T 2 is area under v(t) curve
Slide 29
Displacement integral of velocity lim t 0 t v(t) = area under
v(t) curve note: `area can be positive or negative *Consider v(t)
curve for cart in different situations v t *Net displacement?
Slide 30
Velocity from acceleration curve Similarly, change in velocity
in some time interval is just area enclosed between curve a(t) and
t-axis in that interval. a t TT T2T2 tt a(t)
Slide 31
Summary velocity v = dx/dt = slope of x(t) curve NOT x/t !!
displacement x is v(t)dt = area under v(t) curve NOT vt !! accel. a
= dv/dt = slope of v(t) curve NOT v/t !! change in vel. v is a(t)dt
= area under a(t) curve NOT at !!