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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Lecture 3
Chapter 2
Equations of motion for constant acceleration
Physics I
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Today we are going to discuss:
Chapter 2:
Motion with constant acceleration: Section 2.4 Free fall (gravity): Section 2.5
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Finding Position from a Velocity Graph
The total displacement ∆s is called the “area under the velocity curve.” (the total area enclosed between the t-axis and the velocity curve).
∙ Let’s integrate it: ∙ x ∙
∙
Initial position
Geometrical meaning of an integral is an area
Total displacement
displacement
v(t)
tfiif tandtbetweentvsvunderAreaxx
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Here is the velocity graph of an object that is at the origin (x = 0 m) at t = 0 s.At t = 4.0 s, the object’s position is
A) 20 m
B) 16 m
C) 12 m
D) 8 m
E) 4 m
ConcepTest Position from velocity
Displacement = area under the curve
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Simplifications
Objects are point masses: have mass, no size
In a straight line: one dimension
Point mass
Consider a special, important type of motion:
Acceleration is constant (a = const)
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
The Kinematic Equations of Constant Acceleration
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Velocity equation. Equation 1.(constant acceleration)
0)(, 00
tandtt
vtvaonacceleratidefinitionby o
t
vtva o)(
the velocity is increasing at a constant rate
v0
v
t
atvtv o )( (1)
Velocity equation
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
tvtva o
)(
ff tandtbetweentvsvunderAreaxx 00
v0
v
tO
A
B
CD
Recall Eq (2.11)
vf
t0=0
ABDOADCf AAxx 0
OADCA
ABDA tvvtvxx ff )( 021
00 atvtv o )(
221
00 attvxx f
Position equation. Equation 2 (constant acceleration)
(2)
Position equation
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
We can also combine these two equations so as to eliminate t:
No time equation. Equation 3 (constant acceleration)
It’s useful when time information is not given.
221
00 attvxx f
atvtv o )((3)
No time equation
Position equation
Velocity equation
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
We now have all the equations we need to solve constant-acceleration problems.
Motion at Constant Acceleration (all equations)
221
00 attvxx f
atvtv o )(
(3)
No time equation
Position equation
Velocity equation
(2)
(1)
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
How to solve:– Divide problem into “knowns” and “unknowns”– Determine best equation to solve the problem– Input numbers
Problem Solving
ExampleA plane, taking off from rest, needs to achieve a speedof 28 m/s in order to take off. If the acceleration of theplane is constant at 2 m/s2, what is the minimum lengthof the runway which can be used?
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Exa
mpl
e
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Free
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Freely Falling Objects
Near the surface of the Earth, all objects experience approximately
the same acceleration due to gravity.
All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s2
Air resistance is neglected
One of the most common examples of motion with constant acceleration is freely falling objects.
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The ball is dropped from rest, so its initial velocity is zero. Because the y-axis is pointing upward and the ball is falling downward,its velocity is negative and becomes more and more negative
as it accelerates downward.
You drop a ball. Right after it leaves your hand and before it hits the floor, which of the above plots represents the v vs. t graph for this motion? (Assume your y-axis is pointing up).
ConcepTest Free Fall
v
tA t
B
v
tC
v
tD
yv
v
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Freely Falling Objects
y
g
y
g
a = ‐g
a = g
if then
if then+
++
2
221
00 attvxx f
(3)No time equation
Position equationatvtv o )(
Velocity equation
(2)
(1)
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Example: Ball thrown upward.
A person throws a ball upward into the air with an initial velocity of 10.0 m/s. Calculate
---------------------------------(a) how high it goes, and (b) how long the ball is in the air before it comes back to the hand. (Ignore air resistance.)
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Example
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Example
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
t (s)
Y (m)
0.4 0.8 1.2 1.6 2
5
3
1
v (m/s)
t (s)
Example
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Thank youSee you on Monday
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A cart speeds up toward the origin. What do the position and velocity graphs look like?
ConcepTest 1 Roller Coaster
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Velocity/Acceleration/Position
• 4,5 – negative acceleration, but from 0<t <t4 or t5 – deccelerationbut for t> t4 or t5 – acceleration
a
t
a1>0a2>0a3=0
a4<0a5<0
x0
v
t
12
3
45
v0
t4t5
t
x
t4t5
1 2 3
4
5
V=0
U‐turn
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Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Determining the Sign of the Position, Velocity, and Acceleration