Objectives 2-1 to 2-3
1. Describe the motion of an object relative to a particular frame of reference.
2. Define and calculate displacement (Δx) as the change in position of an object.
3. Differentiate between a vector quantity and a scalar quantity and state which quantities used in kinematics are vector quantities and which are scalar quantities.
4. Define and calculate speed and velocity.
5. Distinguish between speed and velocity.
6. Describe cases where average speed does not equal average velocity.
7. Describe a situation when the velocity is negative.
8. Interpret and analyze a position vs. time graphs for motion.
9. Solve problems involving speed and velocity.
Warm Up
• What do you think of when you hear the terms frame of reference and speed?
Activity
• View the car at the demo table.
• How could you determine the car’s position?
• How could you determine the car’s speed?
• How could you determine the car’s velocity?
Kinematics
• Describes motion while ignoring the agents that caused the motion
2-1 Reference Frames and DisplacementAny measurement of position, distance, or speed must be made with respect to a reference frame.
For example, if you are sitting on a train and someone walks down the aisle, their speed with respect to the train is a few miles per hour, at most. Their speed with respect to the ground is much higher.
Kinematics Problem Solving
• Always choose a reference point.
• The object’s position is its location with respect to a chosen reference point
2-1 Reference Frames and Displacement
We make a distinction between distance and displacement.
Displacement (blue line) is how far the object is from its starting point, regardless of how it got there.
Distance traveled (dashed line) is measured along the actual path.
Where?
• Position—reference point needed
• Distance versus Displacement
Period 1 start here.
Position, Distance, Displacement
• Position—reference point needed
• Distance—no reference point needed
• Displacement—change in position
Position/distance/displacement
• Uses many variables depending on the situation.
• d = “distance” = generic, any direction
• x = in the “x-axis”; sometimes considered the “ground”/”floor” or horizontal motion
• y = in the “y-axis”; sometimes considered the “air/sky” or vertical motion
• s = around the perimeter of a circle
2-1 Reference Frames and Displacement
The displacement is written:
Left:
Displacement is positive.
Right:
Displacement is negative.
Scalar and Vector Period 5 Starts here
• Scalar—quantity that only has a magnitude (distance is an example)
• Vector—quantity that has a magnitude and a direction (displacement is an example)
What is “moving”?
• An object is moving if its position relative to a fixed point is changing.
• Ex. Space shuttle is moving at 30 kilometers per second relative to the sun
or
8 km/s relative to the earth.
2-2 Average Velocity
Speed: how far an object travels in a given time interval
Velocity includes directional information:
(2-1)
Speed and Velocity
• Speed—scalar value (has size only); describes distance traveled per unit of time.
• Velocity—vector value (has size and direction); describes displacement per unit of time.
t
xv
velocity
time
distance
• In racing, is it possible for the car with the greatest speed crossing the finish line to lose the race? Explain.
For both cars, the time elapsed is the distance traveled divided by the average velocity.
Since both cars travel the same distance, the car with the larger average velocity will have the smaller elapsed time.
Constant velocity
• Constant speed and constant direction.
• Does this motion
show constant
speed or
constant
velocity?
Position – Time Graph
Position – Time Graph
run
riseslope
12
12
xx
yyslope
x
yslope
Independent variable “x”
Dependent variable (“y”)
v
d
t
x
yslope
12
12
tt
ddv
if
if
tt
ddv
Formula to find the slope of a line
What would the drop of the slope indicate about the motion?
What is the velocity of this object?
s
mv
0.8
0.24
ss
mmv
0.00.8
0.260.2
smv 0.3
The slope of a position versus time graph is its velocity!
• Slope = -3.0 m/s
• Using the two given data points, the rise can be calculated as -24.0 m (the - sign indicates a drop). The run can be calculated as 8.0 seconds. Thus, the slope is -3.0 m/s.
• The drop indicates the opposite direction which would be backwards or left.
What does this imply about the person’s motion?
distance
time
#6 page 39
• A particle at t1 = -2.0 s is at x1 = 3.4 cm and at t2 = 4.5 s is at x2 = 8.5 cm. What is its average velocity?
• Can you calculate its average speed from these data?
#9 page 39
• A person jogs eight complete laps around a quarter-mile track in a total time of 12.5 min?
• Calculate (a) the average speed and (b) the average velocity, in m/s.
2-3 Instantaneous Velocity
The instantaneous velocity is the average velocity, in the limit as the time interval becomes infinitesimally short.
These graphs show (a) constant velocity and (b) varying velocity.
(2-3)
Given the curve:
x
f(x)
x+h
f(x+h)
sec
f x h f xm
h
tan
h 0
f x h f xm lim
h
NUMERICALLY
t (hours) 0 1 3 4 5 7h (meters) 18.2 17.1 18.9 23.2 25.1 25.4
A. Find the average velocity over the interval 1 < t < 3.
ave sec
18.9 17.1v m 0.9
3 1
B. Using appropriate units, explain the meaning of your answer.
0.9 represents the average meters per hour of a particle from t = 1 hour to t = 3 hours
NUMERICALLY
t (hours) 0 1 3 4 5 7h (meters) 18.2 17.1 18.9 23.2 25.1 25.4
A. Find the average velocity over the interval 0 < t < 4
ave sec
23.2 18.2v m 1.25
4 0
B. Using appropriate units, explain the meaning of your answer.
1.25 represents the average meters per hour of a particle from t = 0 to t = 4 hours
NUMERICALLY
t (hours) 0 1 3 4 5 7h (meters) 18.2 17.1 18.9 23.2 25.1 25.4
A. Estimate the velocity at t = 5.
Note: velocity implies INSTANTANEOUS velocity
sec
25.4 23.2v m 0.733
7 4
B. Using appropriate units, explain the meaning of your answer.
The velocity is approximately 0.733 meters per hour at t = 5 hours.
GRAPHICALLY
Find the average rate of change of f(x) on [-2, 2]
sec
3 1m 1
2 2
Estimate the instantaneous rate of change of f(x) at x = 0
sec
1.7 1.7f ' x m 1.7
1 1
Find the average velocity of the ship in the first two hours
ave sec
6.5 0v m 3.25
2 0
Estimate the velocity of the ship after 75 minutes
sec
5.8 4.2v m 3.2
1.5 1
Closure
1.When did the person stop moving?
2.What happened at 9 minutes?
3.When did he have the greatest speed?
Step by Step Graph
-20
-15
-10
-5
0
5
10
15
0 2 4 6 8 10 12 14
time (min)
ste
ps
(m
)
Homework
Chapter 2 Problems: 7, 11, 14
