which car/s is/are undergoing an acceleration? which car experiences the greatest acceleration?
TRANSCRIPT
Which car/s is/are undergoing an acceleration?
Which car experiences the greatest acceleration?
Match a Graph• Consider the position-time graphs
below. Each one of the 3 lines on the position-time graph corresponds to the motion of one of the 3 cars. Match the appropriate line to the particular color of car.
Section 1: Position vs Time Graphs
Position vs Time Graphs• position (x) vs time (t) graphs plot
position as a function of time
• the x-t graph of a constantly moving object is linear
• If an object is staying in the same position for a time interval, then it’s…
• at rest.• stopped.• not moving.• still.
• Represented by a horizontal line on a position vs time graph
• velocity- displacement traveled in a certain amount of time; a vector quantity
• can be constant or average
• constant velocity- velocity that does not change
Constant Velocity
Constant Velocity• Questions:
1. What is slope?
2. What is the formula for slope?
3. Use the slope formula to determine the slope of the line.
4. What do you notice?
Constant Velocity• Conclusions:
1. the slope of a position vs time graph is the velocity
2. m = rise = ∆y = displacement = v
run ∆x time
• Questions:
1. Where does the graph have positive slope?
2. What does this positive slope mean?3. Where does the graph have negative
slope?4. What does this negative slope mean?5. Is there any place(s) on the graph where
the object is not moving?6. How do you know?7. What is the slope of the graph at this
section?
Average Speed vs Average Velocity• Velocity is not the same as speed.
• average velocity- the change in position divided by the time it took for that change to occur; a ratio of an object’s change in position (displacement) to the time it takes for that change to occur
vavg = distance traveled time of travel
vavg = ∆x ∆t
Speed or Velocity?• speed & velocity are different
quantities• speed is the rate distance covered
over time• speed is a scalar quantity• velocity is the rate displacement
traveled over time• velocity is a vector quantity• speed & velocity equal magnitudes
when an object does not change direction
Average Velocity • Consider a car moving with a constant
velocity of +10 m/s for 5 seconds. The diagram below depicts such a motion.
Average Velocity• The position-time graph would look like
the graph below:
Average Velocity• Now consider a car moving at a
constant velocity of +5 m/s for 5 seconds, abruptly stopping, & then remaining at rest (v = 0 m/s) for 5 seconds.
How would this graph look?
Average Velocity• Check your answer…
Instantaneous Speed• speed at any given point in time
Can I use one point to find velocity?
Important Note: DO NOT use one point to calculate velocity on a position-time graph!!!
Example 2—Drawing Graphs from Graphs
• Use the position vs time below to draw a velocity vs time graph.
SolutionVelocity vs Time
Vel
ocity
(m
/s)
Time (s)654321
-12
-8
-4
0
12
8
4
Position vs Time Graphs• the x-t graph of a constantly moving
object is linear
Position vs Time Graphs• the slope of an x-t graph is rise
(position) over run (time)
• for an x-t graph,
m = ∆x ∆t
Position vs Time Graphs• the change in position of an object is
displacement, ∆x
• the rate change in position of an object over an interval of time (the time it takes for the change to occur) is called average velocity, vavg
vavg = ∆x ∆t
Position vs Time Graphs
Position vs Time Graph
Position vs Time Graphs• the slope of an x-t graph describes
the velocity of an object in motion
Slow, Rightward(+)Constant Velocity
Fast, Rightward(+)Constant Velocity
Position vs Time Graphs• the slope of an x-t graph describes
the velocity of an object in motionSlow, Leftward(-)Constant Velocity
Fast, Leftward(-)Constant Velocity
Position vs Time Graphs
Position vs Time Graphs
Quadrant I
Quadrant IV
Position vs Time Graphs
Section 2: Velocity vs Time Graphs
Velocity vs Time Graphs• a velocity (v) vs time (t) graph plots
velocity as a function of time
• the v-t graph of a constantly moving object is a horizontal line
• remember, horizontal lines have zero slope
Velocity vs Time Graphs• the v-t graph of an object moving at a
changing rate is a diagonal line
• the velocity is changing by a constant rate
Constant, Uniform Acceleration• Bessie took a trip on her bicycle. The
data of Bessie’s bike ride below, along with its accompanying graph, is shown. Use the graph to answer the questions that follow:
Constant, Uniform Acceleration• Questions:
1. What do you notice about this position vs time graph?
2. What does that mean?
Constant, Uniform Acceleration• Now, here is the data, along with the
resulting velocity vs time graph, of the same bike ride.
Constant, Uniform Acceleration• Questions:
1.What can you say about the slope of this graph?
2.What does that mean?
Constant, Uniform Acceleration• Conclusions:
– If an object has constant velocity, then that object has zero acceleration.
– Constant velocity Δv equals zero.– The graph of zero acceleration is a
horizontal line.– All horizontal lines have zero slope
because Δy equals zero.
Constant, Uniform Acceleration
Velocity vs Time Graphs• the slope of a v-t graph describes
changing velocity over time
m = ∆v ∆t
Velocity vs Time Graphs• the rate at which velocity changes
over an interval time is known as acceleration, aavg
• mathematically,
aavg = ∆v ∆t
Velocity vs Time Graphs• the slope of a v-t graph describes the
acceleration of the object in motion
Positive VelocityPositive Acceleration
Velocity vs Time Graphs• units of acceleration: m/s/s, m/s2
• acceleration is a vector quantity
• acceleration describes a change in velocity in the positive direction
• acceleration also describes a change in direction
• deceleration (negative acceleration) describes a change in velocity in the opposite direction
Velocity vs Time Graphs
Velocity vs Time Graphs
Velocity vs Time Graphs
Velocity vs Time Graphs
Section 2a: Constant Acceleration
Example 2—Drawing Graphs from Graphs
• Use the position vs time below to draw a velocity vs time graph.
Solution Velocity vs Time
Vel
ocity
(m
/s)
Time (s)654321
-12
-8
-4
0
12
8
4
SolutionAcceleration vs Time
Acc
eler
atio
n (m
/s2 )
Time (s)654321
-12
-8
-4
0
12
8
4
Example 3—Drawing Graphs from Graphs
Constant, Uniform Acceleration• velocity changes by the same rate
• ex: a ball rolling down a ramp
Constant, Uniform Acceleration
Constant, Uniform Acceleration• Questions:
1. What is slope?
2. What is the formula for slope?
3. Use the slope formula to determine the slope of the line.
4. What do you notice?
Constant, Uniform Acceleration• Conclusions:
1. the slope of a velocity vs time graph is acceleration
2. m = rise = ∆y = change in velocity = a
run ∆x time
Constant, Uniform Acceleration
Section 3: Using Tangent Lines
Using Tangent Lines• What does the following graph show?
0
50
100
150
200
0 5 10
Time (hours)
Dis
tan
ce (
Mile
s)
Draw a straight line that is tangent to the curve.
What does tangent mean?
Position vs Time
Time (s)
Pos
ition
(m
)
Using Tangent Lines• What does this graph show?
0
20
40
60
80
100
0 5 10
Time (hours)
Dis
tan
ce (
Mile
s)
Constant Acceleration vs Constant Velocity
• position vs time graph with increasing speed & its resulting velocity vs time graph:
positive slope positive acceleration
• position vs time graph with decreasing speed & its resulting velocity vs time graph:
negative slope negative acceleration
Animations
Animations
Animations
Example: Determine what is happening at each section
constant velocity
acceleration
increasing accelerationdeceleration
Section 4: Lines That Intersect
Lines That Intersect• Sometimes, a problem will ask you to
compare/contrast the motion of 2 objects.
• To do this, use the information provided in the problem to construct a position vs time graph.
• Then, use the graph to compare/contrast the 2 objects’ motions.
Lines That Intersect
What do the 2 cars have in common?
Do they ever share the same velocity?
Example 2—100 meter dash• Harry gives Sam a 30. m head start in
the 100. m dash. Harry can run at 10.0 m/s, while Sam only runs at 6.0 m/s.
Example• Does Harry win?
Solution• Draw a position vs time graph for a
100. m dash, plotting both runners.Position vs Time
Position (m)
Time (s)
100
50
5 10
Example• If Harry wins, at what time & place
does he catch up to Sam?
Solution• Now, you can determine the equation
of each line.
• Remember, the equation of a line is
• Since we are using position, time, & velocity, the equation becomes
y = mx + b
Solution
x = vt + b
Solution• For Harry…
• Where, vH = Harry’s velocity
• bH = where Harry begins
xH = vHtH + bH
Solution• So…
xH = 10t
Solution• For Sam…
• Where, vS = Sam’s velocity
• bS = where Sam begins
xS = vStS + bS
Solution• So…
xS = 6t + 30
Solution• Since you are looking for the point
where Harry & Sam pass each other, that means the coordinates (tH, xH) & (tS, xS) are equal.
• So, set the 2 equations for each line equal to each other & solve for t.
Solution
10t = 6t + 30 4t = 30 t = 7.5 sec
Solution• Now, that you know at what time
Harry catches up to Sam, you can calculate at what place this happens.
• Plug the value for time (7.5 sec) into the equation & solve for x.
xH = 10t = (10.0 m/s) (7.5 s) xH = 75 m
Section 5: Displacement from V-T Graphs
Displacement from a Velocity vs Time Graph
ve
loci
ty
time t1 t2
v
area = l* w = v (t2 – t1)
= ∆x
• in the given v-t graph, the area under the graph is that for a rectangle
Displacement from a Velocity vs Time Graph
ve
loci
ty
time t1 t2
v
∆x
• in the given v-t graph, the area under the graph is that for a rectangle
