Ch02 Motion Along a Straight Line -...
Transcript of Ch02 Motion Along a Straight Line -...
Chapter 2
Motion Along A Straight Line
2 Motion Along A Straight Line
4 October 2018 2 PHY101 Physics I © Dr.Cem Özdoğan
4 October 2018 3 PHY101 Physics I © Dr.Cem Özdoğan
Announcement
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2-2 Motion
• We find moving objects all around us.
• One purpose of physics is to study the motion of objects
• how fast they move
• how far they move in a given amount of time...
• The classification and comparison of motions: kinematics.
• In this chapter;
1.We study the basic physics of motion where the object moves along a
single axis. (one-dimensional motion)
2.Forces cause motion. We will find out, as a result of application of
force, if the objects speed up, slow down, or maintain the same rate.
3.The moving object here will be considered as a particle.
• If we deal with a stiff, extended object, we will assume that all
particles on the body move in the same fashion.
• We will study the motion of a particle, which will represent the entire
body.
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2-3 Position and Displacement
• To locate an object = to find its
position relative to some reference
point (origin).
• A change from position x1 to position x2
is called a displacement Δx: Δ (delta): change in quantity
• Displacement is a vector quantity (has both a direction and a
magnitude).
• An object’s displacement is -4 m means that the object has moved
towards decreasing x-axis by 4 m. The direction of motion, here, is
toward decreasing x.
• From x = 5 m to x = 12 m: ∆x = 7 m (positive direction)
• From x = 5 m to x = 1 m: ∆x = -4 m (negative direction)
• From x = 5 m to x = 200 m to x = 5 m: ∆x = 0 m
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2-4 Average Velocity and Average Speed
How Fast? Average Velocity • How to describe position? Plot x as a function of t : graph of x(t)
• Ex: Straight-line motion of an animal: How fast does this animal
move? Average Velocity:
Vavg: The slope of the straight
line that connects two particular
points on the x(t) curve.
Vavg has both magnitude and
direction (it is another vector
quantity)
Unit: m/s
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How Fast? Average Speed • Calculation of the average velocity between t =1s and t =4s as the
slope of the line that connects the points.
Start
• Positive slope means positive
average velocity
• Negative slope means negative
average velocity
2-4 Average Velocity and Average Speed
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How Fast? Average Speed Q: What is the difference btw Ave velocity ans Ave speed?
A: Average velocity involves the particle’s displacement x, the average
speed involves the total distance covered independent of direction!
• Because average speed does not include direction, it
lacks any algebraic sign! Average velocity = -3 m/s;
average speed = 3 m/s
Average speed is always
positive (no direction)
2-4 Average Velocity and Average Speed
AK Lecture Notes
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• Example: You drive a truck along a straight road for 8.4 km at 70
km/h, at which point the truck runs out of gasoline and stops.
• Over the next 30 min, you walk another 2.0 km farther along the road
to a gasoline station.
(a)What is your overall displacement from the beginning to your
arrival at the station?
(b)What is the time interval t from the beginning of your drive to your
arrival at the station?
2-4 Average Velocity and Average Speed
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(c)What is your average velocity vavg from
the beginning of your drive to your arrival
at the station? Find it both numerically and
graphically.
(d)Suppose that to pump the gasoline, pay for it, and walk back to the
truck takes you another 45 min. What is your average speed from the
beginning of your drive to your return to the truck with the gasoline?
2-4 Average Velocity and Average Speed
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2-5 Instantaneous Velocity and Speed
• “How fast” more commonly refers to how fast a particle is moving at a given instant
• At a single moment in time,
obtained from average
velocity by shrinking ∆t.
• v: the rate at which position x
is changing with time at a
given instant (the slope of the
position-time curve at the
point representing that
instant).
• Speed is the magnitude of velocity (velocity that has been stripped of
any indication of direction).
AK Lecture Notes
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Example:
The slope of
x(t), and so also
the velocity v, is
zero from 0 to 1
s, and from 9s
on.
During the
interval bc, the
slope is constant
and nonzero, so
the cab moves
with constant
velocity (4 m/s).
2-5 Instantaneous Velocity and Speed
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2-6 Acceleration
• Average acceleration is the change of velocity
over the change of time.
• Instantaneous acceleration is the rate at
which its velocity is changing at that instant.
• In terms of the position function, the
acceleration can be defined as:
• The SI unit for acceleration is m/s2.
• Acceleration has both magnitude and direction (vector
quantity).
• If a particle has the same sign for velocity and acceleration,
then that particle is speeding up.
• If a particle has opposite signs for the velocity and
acceleration, then the particle is slowing down.
Example: If a car with velocity v = -25 m/s is braked to a stop
in 5.0 s, then a = + 5.0 m/s2. Acceleration is positive, but speed
has decreased.
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(a)Motion diagram for a car moving at constant velocity (zero acceleration).
(b)Motion diagram for a car whose constant acceleration is in the direction of
its velocity. The velocity vector at each instant is indicated by a red arrow,
and the constant acceleration is indicated by a violet arrow.
(c)Motion diagram for a car whose constant acceleration is in the direction
opposite the velocity at each instant.
2-6 Acceleration
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• Our bodies often react to accelerations but not to
velocities. A fast car often does not bother the rider,
but a sudden brake is felt strongly by the rider. This is
common in amusement car rides, where the rides
change velocities quickly to thrill the riders.
• The magnitude of acceleration falling near the Earth’s
surface is 9.8 m/s2, and is often referred to as g.
In a rocket sled, which undergoes sudden change in velocities.
2-6 Acceleration
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Example
Continued
When acceleration is 0
(e.g. interval bc) velocity
is constant.
When acceleration is
positive (ab) upward
velocity increases.
When acceleration is
negative (cd) upward
velocity decreases.
Steeper slope of the
velocity-time graph
indicates a larger
magnitude of acceleration:
the cab stops in half the
time it takes to get up to
speed.
2-6 Acceleration
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Example:
2-6 Acceleration
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2-7 Constant Acceleration: A Special Case
• In many cases acceleration is constant, or nearly so.
• When the acceleration is constant, its average and instantaneous
acceleration values are the same.
(1) Here, velocity at t=0 is vo.
Average = ((initial) + (final)) / 2:
finally leading to (2)
• Eliminating t from the Equations (1) and (2):
5 special equations
(4)
(5)
(3)
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2-7 Constant Acceleration: A Special Case
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• Integrating constant
acceleration graph for a
fixed time duration yields
values for velocity graph
during that time.
• Similarly, integrating
velocity graph will yield
values for position graph.
• Note that constant
acceleration means a
velocity with a constant
slope, and a position with
varying slope (unless a = 0).
2-7 Constant Acceleration: A Special Case
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Example:
2-7 Constant Acceleration: A Special Case
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2-9 Free-fall Acceleration
• Free-fall acceleration is the rate at which
an object accelerates downward in the
absence of air resistance.
• In this case objects close to the Earth’s
surface fall towards the Earth’s surface
with no external forces acting on them
except for their weight.
• Use the constant acceleration model with
“a” replaced by “-g”, where g = 9.8 m/s2
for motion close to the Earth’s surface.
• The equations of motion in Table 2-1
apply to objects in free-fall near
Earth's surface. AK Lecture Notes
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2-9 Free-fall Acceleration
• In vacuum, a feather and an apple will fall
at the same rate. Independent of the
properties of the object (mass, density,
shape, see Figure)
AK Lecture Notes
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Example:
2-9 Free-fall Acceleration
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Example:
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2 Solved Problems
1. During a hard sneeze, your eyes might shut for 0.50 s. If you are driving a car at
90 km/h during such a sneeze, how far does the car move during that time?
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2. An electron moving along the x-axis has a position given by x=16te-t m, where t
is in seconds. How far is the electron from the origin when it momentarily stops?
2 Solved Problems
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3.The position of a particle moving along an axis is given by x=12t2-2t3, where x is in
meters and t is in seconds. Determine (a) the position, (b) the velocity, and (c) the
acceleration of the particle at t=3.0 s . (d) What is the maximum positive coordinate
reached by the particle and (e) at what time is it reached? (f) What is the maximum
positive velocity reached by the particle and (g) at what time is it reached? (h) What
is the acceleration of the particle at the instant the particle is not moving (other than at
t=0)? (i) Determine the average velocity of the particle between t=0 and t=3 s.
2 Solved Problems
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3. (Continued) The position of a particle moving along an axis is given by x=12t2-2t3,
where x is in meters and t is in seconds. Determine (a) the position, (b) the velocity,
and (c) the acceleration of the particle at t=3.0 s . (d) What is the maximum positive
coordinate reached by the particle and (e) at what time is it reached? (f) What is the
maximum positive velocity reached by the particle and (g) at what time is it reached?
(h) What is the acceleration of the particle at the instant the particle is not moving
(other than at t=0)? (i) Determine the average velocity of the particle between t=0
and t=3 s.
2 Solved Problems
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4. An electron with an initial velocity v0 = 1.50x105 m/s
enters a region of length L=1.00 cm where it is
electrically accelerated (see Figure). It emerges with
v = 5.70x106 m/s . What is its acceleration, assumed
constant?
2 Solved Problems
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5. The brakes on your car can slow you at a rate of 5.2 m/s2. (a) If you are going
137 km/h and suddenly see a state trooper, what is the minimum time in which
you can get your car under the 90 km/h speed limit? (b) Graph versus and versus
for such a slowing.
2 Solved Problems
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6. (a) With what speed must a ball be thrown vertically from ground level to rise to a
maximum height of 50 m? (b) How long will it be in the air? (c) Sketch graphs of
y, v, and a versus t for the ball. On the first two graphs, indicate the time at which
50 m is reached.
2 Solved Problems
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6. (a) With what speed must a ball be thrown vertically from ground level to rise to a
maximum height of 50 m? (b) How long will it be in the air? (c) Sketch graphs of
y, v, and a versus t for the ball. On the first two graphs, indicate the time at which
50 m is reached.
2 Solved Problems
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7. To test the quality of a tennis ball, you drop it onto the floor from a height of
4.00 m. It rebounds to a height of 2.00 m. If the ball is in contact with the floor for
12.0 ms, (a) what is the magnitude of its average acceleration during that contact
and (b) is the average acceleration up or down?
2 Solved Problems
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Position Relative to origin
Positive and negative
directions
Displacement Change in position (vector)
Average Speed Distance traveled / time
Average Velocity Displacement / time (vector)
Eq. (2-1)
Eq. (2-2) Eq. (2-3)
2 Summary
Instantaneous Velocity At a moment in time
Speed is its magnitude
Average Acceleration Ratio of change in velocity to
change in time
Eq. (2-4) Eq. (2-7)
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Constant Acceleration
Includes free-fall, where
a = -g along the vertical axis
Instantaneous Acceleration
First derivative of velocity
Second derivative of position
Eq. (2-8) Tab. (2-1)
2 Summary
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Additional Materials
2 Motion Along A Straight Line
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2-10 Graphical Integration in Motion Analysis
Integrating acceleration:
Starting from
we obtain
(vo= velocity at time t=0, and v1= velocity at
time t = t1).
Note that
Integrating velocity:
Similarly, we obtain
(xo= position at time t = 0, and x1 = position
at time t=t1), and
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Example:
2-10 Graphical Integration in Motion Analysis
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2-1_4 Position, Displacement, and Average Velocity
• 2.01 Identify that if all parts
of an object move in the same
direction at the same rate, we
can treat it as a (point-like)
particle.
• 2.02 Identify that the position
of a particle is its location on a
scaled axis.
• 2.03 Apply the relationship
between a particle's
displacement and its initial
and final positions.
• 2.04 Apply the relationship
between a particle's average
velocity, its displacement, and
the time interval.
• 2.05 Apply the relationship
between a particle's average
speed, the total distance it
moves, and the time interval.
• 2.06 Given a graph of a
particle's position versus time,
determine the average velocity
between two times.
Learning Objectives
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2-5 Instantaneous Velocity and Speed
• 2.07 Given a particle's
position as a function of time,
calculate the instantaneous
velocity for any particular
time.
• 2.08 Given a graph of a
particle's position versus time,
determine the instantaneous
velocity for any particular
time.
• 2.09 Identify speed as the
magnitude of instantaneous
velocity.
Learning Objectives
4 October 2018 42 PHY101 Physics I © Dr.Cem Özdoğan
2-6 Acceleration
• 2.10 Apply the relationship
between a particle's average
acceleration, its change in
velocity, and the time interval
for that change.
• 2.11 Given a particle's
velocity as a function of time,
calculate the instantaneous
acceleration for any particular
time.
• 2.12 Given a graph of a
particle's velocity versus time,
determine the instantaneous
acceleration for any particular
time and the average
acceleration between any two
particular times.
Learning Objectives
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• 2.13 For constant
acceleration, apply the
relationships between
position, velocity,
acceleration, and elapsed time
(Table 2-1).
• 2.14 Calculate a particle's
change in velocity by
integrating its acceleration
function with respect to time.
• 2.15 Calculate a particle's
change in position by
integrating its velocity
function with respect to time.
Learning Objectives
2-7 Constant Acceleration
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2.16 Identify that if a particle
is in free flight (whether
upward or downward) and if
we can neglect the effects of
air on its motion, the particle
has a constant downward
acceleration with a magnitude
g that we take to be 9.8m/s2.
2.17 Apply the constant
acceleration equations (Table
2-1) to free-fall motion.
Learning Objectives
Figure 2-12
2-9 Free-Fall Acceleration
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• 2.18 Determine a particle's
change in velocity by
graphical integration on a
graph of acceleration versus
time.
• 2.19 Determine a particle's
change in position by
graphical integration on a
graph of velocity versus time.
Learning Objectives
2-10 Graphical Integration in Motion Analysis