equations of motion for constant acceleration
TRANSCRIPT
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
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
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
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
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)
Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
The Kinematic Equations of Constant Acceleration
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
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
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
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)
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?
Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Exa
mpl
e
Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Free
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.
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
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)
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.)
Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Example
Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Example
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
Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Thank youSee you on Monday
A cart speeds up toward the origin. What do the position and velocity graphs look like?
ConcepTest 1 Roller Coaster
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
Department of Physics and Applied PhysicsPHYS.1410 Lecture 3 Danylov
Determining the Sign of the Position, Velocity, and Acceleration