Lecture 2:One Dimensional Motion
2
Scalars and Vectors
Scalar – a numerical value. May be positive or negative. Examples: temperature, speed, height
Vector – a quantity with both magnitude and direction. Examples: displacement (e.g., 10 feet north), force, magnetic
field
“position” is obviously a vector.
Example: when looking at a 2-dimensional map “2 miles South”
Note: this implicitly has 2 numbers in it: “0 miles East”
“position” corresponds to “displacement”, a vector.
length of travel is “distance”, a scaler
Position, Distance, and DisplacementBefore describing motion, you must set up a coordinate system – define an origin and a positive direction.
The distance is the total length of travel; if you drive from your house to the grocery store and back, you have covered a distance of 8.6 mi. Your net displacement is zero.
Position, Distance, and Displacement
If you drive from your house to the grocery store and then to your friend’s house, your displacement is -2.1 mi and the distance you have traveled is 10.7 mi.
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
Walking the Dog
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.
Walking the Dog
Follow-up: have you and your dog traveled the same distance?
Average SpeedThe average speed is defined as the distance traveled divided by the time the trip took:
Average speed = distance / elapsed time
If you drive from your house to the grocery store and then to your friend’s house, your displacement is -2.1 mi and the distance you have traveled is 10.7 mi. The trip takes 30 minutes (with traffic and lights).What is your average speed?
10.7 miles
0.5 hourssav =
= 21.4 miles/hr
Average VelocityAverage velocity = displacement / elapsed time
If you return to your starting point, your average velocity is zero.
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, you affectionately
pat your dog’s head.
Which statement correctly describes
your average speed and velocity
relative to that of your dog, on the
trip from home to the park?
a) Average speed and average velocity were both different
b) Average speed was the same, but average velocity was different
c) Average speed was different, but average velocity was the same
d) Average speed and average velocity were the same
Walking the Dog II
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, you affectionately
pat your dog’s head.
Which statement correctly describes
your average speed and velocity
relative to that of your dog, on the
trip from home to the park?
a) Average speed and average velocity were both different
b) Average speed was the same, but average velocity was different
c) Average speed was different, but average velocity was the same
d) Average speed and average velocity were the same
Walking the Dog II
Your dog’s many side trips mean that he travelled more distance than you
in the same amount of time, so his average speed must have been greater.
However, his net displacement was the same (starting and ending at the
same place), so his average velocity must have been the same.
Averaging Speed
Is the average speed of the red car a) 40.0 mi/h, b) more than 40.0 mi/h, or c) less than 40.0 mi/h?
Hint: is t1 = t2?
Position vs. Time
Consider this motion sequence, plotted here in one-dimension...
...and here as an x-t graph
average velocity
Instantaneous Velocity
The instantaneous velocity is tangent to the curve.
Evaluating the average velocity over a shorter and shorter period of time, one approaches the “instantaneous velocity”.
Graphical Interpretation of Average and Instantaneous Velocity
Position v. Time IPosition v. Time I
t
x
The graph of position versus
time for a car is given below.
What can you say about the
velocity of the car over time?
a) it speeds up all the time
b) it slows down all the time
c) it moves at constant velocity
d) sometimes it speeds up and
sometimes it slows down
e) not really sure
t
x
The graph of position versus
time for a car is given below.
What can you say about the
velocity of the car over time?
The car moves at a constant velocity
because the x vs. t plot shows a straight
line. The slope of a straight line is
constant. Remember that the slope of x
vs. t is the velocity!
a) it speeds up all the time
b) it slows down all the time
c) it moves at constant velocity
d) sometimes it speeds up and
sometimes it slows down
e) not really sure
Position v. Time I
t
x
a) it speeds up all the time
b) it slows down all the time
c) it moves at constant velocity
d) sometimes it speeds up and
sometimes it slows down
e) not really sure
The graph of position vs.
time for a car is given below.
What can you say about the
velocity of the car over
time?
Position v. Time IIPosition v. Time II
a) it speeds up all the time
b) it slows down all the time
c) it moves at constant velocity
d) sometimes it speeds up and
sometimes it slows down
e) not really sure
x
The graph of position vs.
time for a car is given below.
What can you say about the
velocity of the car over
time?
The car slows down all the time because the slope of the x vs. t graph is diminishing as time goes on. Remember that the slope of x vs. t is the velocity! At large t, the value of the position x does not change, indicating that the car must be at rest.
t
Position v. Time IIPosition v. Time II
Acceleration
Average acceleration:
Instantaneous acceleration:
Graphical Interpretation of Average and Instantaneous Acceleration
Sign of Acceleration
Speed Decreasing
Does deceleration mean “negative acceleration”?
No. It means “decreasing speed”.
Acceleration
Acceleration (increasing speed) and deceleration (decreasing speed) should not be confused with the direction (or sign) of velocity and acceleration:
Graphical Interpretation of Average and Instantaneous Acceleration
Constant Acceleration
If the acceleration is constant, the velocity changes linearly with time:
Average velocity:
Propeller carPropeller carv
ta:
v
tb:
v
tc:
v
td:
Which of the above plots
represents the v vs. t
graph for the motion of
the propeller car after it
was pushed?
e: Not Sure
Propeller carPropeller carv
ta:
v
tb:
v
tc:
v
td:
Which of the above plots
represents the v vs. t
graph for the motion of
the propeller car after it
was pushed?
The car has a negative initial velocity, but a constant positive acceleration. So the plot should start at a negative value, and increase linearly with increasing time.
e: Not Sure
Motion with Constant AccelerationPosition as a function of time
Motion with Constant Acceleration
The relationship between position and time follows a characteristic curve.
My guess: a ~ 7 m/s2
Propeller car Propeller car displacementdisplacement
Which of the displayed plots represents the x vs. t graph for the motion of the propeller car after it was pushed?
x
t
Propeller car Propeller car displacementdisplacement
Which of the displayed plots represents the x vs. t graph for the motion of the propeller car after it was pushed?
The car starts at zero position, with a negative initial velocity, so the displacement is growing in the
negative direction. The slope is changing linearly with time to become more positive. It should be a smooth
change, go through a minimum, and then change more and more quickly with increasing timex = v0t +
1/2 at2
x
t
Motion with Constant AccelerationVelocity as a function of position
Stopping Distance
1/2 Stopping Distance
3/4 Stopping Distance
Tip for save driving: if you double your speed, what happens to your stopping distance?
v = 0.7 v0
deaccelerating with constant a, take final velocity = 0,
find (x-x0) = “stopping distance”
Hit the Brakes!
Freely Falling ObjectsFree fall from rest:
Free fall is the motion of an object subject only to the influence of gravity. The acceleration due to gravity is a constant, g.
g = 9.8 m/s2
For free falling objects, assuming your y axis is
pointing up, a = -g = -9.8 m/s2
- Clickers, everyday, in class.
- Assignment 1 on MasteringPhysics. Due Friday, August 30, at midnight!
- Reading, for next class (Chapter 2, 3.1-3.6)
- When you exit, please use the REAR doors!