Kinematics in One Dimension -...
Transcript of Kinematics in One Dimension -...
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Lecture 2
Chapter 1 and 2
Kinematics in One Dimension
Physics I
ResponseWare App instead of a clicker
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Today we are going to discuss basic kinematic stuff:
Distance/Displacement: Section 1.3
Speed/Average velocity: Section 1.4 Instantaneous velocity: Section 2.2
Average Acceleration: Section 1.5 Instantaneous Acceleration: Section 2.7
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Mechanics
There are three branches of Mechanics:Kinematics Motion Forces Statics Motion ForcesDynamics Motion Forces
Kinematics describes motion of objects we are not interested in reasons (forces) of a motion
Motion
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Kinematics
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Frames of Reference
-5 5
-5
5 y-axis
x-axis
Before starting solving any problem, we have to define a coordinate system (a frame of reference) to describe position and motion of an object
In this class, we will base problems in a Cartesian coordinate system. We will have 1D and 2D problems
We have freedom to choose direction of an axis. For 1 dimensional (1D) motion (motion in a straight line), it’s better to align the x-axis along a
motion direction. For falling bodies, we tend to describe position using the y-axis
origin
1 2 3 4 5
Positive direction+x
Negative direction-x
x
One dimensional (1D) problem Two dimensional (2D) problem
origin
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Distance vs. Displacement
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Distance vs. Displacement
Distance (scalar): the total path length traveled by an object
Displacement (vector): how far an object is from its starting point
What would be your displacement after a complete roller coaster?
ConcepTest 1 Roller Coaster
You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement?
A) yes
B) no
Yes, you have the same displacement. Because you and your dog had the same initial position and the same final position, then you have (by definition) the same displacement.
ConcepTest 2 Walking the Dog
Follow-up: have you and your dog traveled the same distance?
Follow-up: how would you measure displacement in your car?
Does the odometer in a car
measure distance or
displacement?
A) distance
B) displacement
C) both
ConcepTest 3 Odometer
If you go on a long trip and then return home, your odometer does not measure zero, but it records the total miles that you traveled. That means the odometer records distance.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Distance vs. Displacement (1D)Distance is a scalar
Displacement is a vector– A vector has both magnitude and direction (or sign in 1-D)
Displacement =x2- x1=+40 m
Distance =
20 40 60 70
x2
x
x1
70+30 =100 m
Displacement = final position – initial position
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Distance vs. Displacement (1D)
x1 x2
Distance = 20 m
x1
Distance = 20 m
Displacement =negative
Displacement =
x1
30-10= + 20 m
x2
10-30= -20 m
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Speed and Velocity
Even Hollywood feels that there is a difference between these two terms
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Average Speed and Average Velocity
elapsed time travelleddistance
speedaverage
(Speed: Distance traveled per unit time interval)
Speed is a scalar
Velocity is a vector
(Velocity: Displacement of an object per unit time interval)
elapsed timentdisplaceme
velocityaverage
Does the speedometer in a car
measure velocity or speed?
A) velocity
B) speed
C) both
D) neither
The speedometer clearly measures speed, not velocity. Velocity is a vector (depends on direction), but the speedometer does not care what direction you are traveling.
ConcepTest 4 Speedometer
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Graphs: Average velocity
0 1 2 3 4 5 60
5
10
15
20po
sitio
n (m
)
time (s)
∆t
∆x
t1 t2
x1
x2
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Instantaneous VelocityAverage velocity does not tell the whole story…we also need
If you watch a car’s speedometer, at any instant of time, the speedometer tells you how fast the car is going at that instant.
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Instantaneous velocity
0 1 2 3 4 5 60
5
10
15
20po
sitio
n (m
)
time (s)
t1
goes to 0
Instantaneous velocity
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Instantaneous velocity
Graphically, instantaneous velocity is the slope of the x vs t plot at a single point
Mathematically, the instantaneous velocity is the derivative of the position function
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Finding instantaneous velocity from position graphically
.
>0
Turning point
Moves forward Moves backward
0
v positive
It flies back.v4 negative
v1 (slow speed)v2 (max speed)
v3=0 (turning point)
Steeper slope ≡ higher velocityGentler slope ≡ lower velocity
Which one of the following x-vs-t graphs could be a
reasonable representation of the motion of a baton in
a relay race being passed from one runner to the next?
ConcepTest 5 Relay baton
End of class
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 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).
The displacement is the shaded area
∆
Example
∙ Let’s integrate it: ∙ x ∙
∙
Initial position
Geometrical meaning of an integral is an area
Total displacement
displacement
v(t)
t
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 2 Danylov
Acceleration
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
AccelerationVelocity can also change with time: acceleration
elapsed time velocityof change
onacceleratiaverage
Speeding up: acceleration
Instantaneous acceleration
If we are given x(t), we can find both velocity v(t) and acceleration a(t) as a function of time
Slowing down: deceleration
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
A particle is moving in a straight line so that its position is given by the relation
x = (2 m/s2)t2 + (3 m). Calculate
(a) its average acceleration during the timeinterval from t1 = 1 s to t2 = 2 s,
(b) its instantaneous acceleration as afunction of time.
Example 2-7: Acceleration given x(t).
Department of Physics and Applied PhysicsPHYS.1410 Lecture 2 Danylov
Thank youSee you on Wednesday