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When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. For this question, many students will say the book is at rest, while others may say that Earth is moving so the book is moving as well. Students will sometimes say the molecules are moving so the book is moving. The point of the question is to lead them to the concept of a frame of reference.
Students sometimes just subtract the smaller from the larger number instead of the initial position from the final position.
These same sign conventions will apply to velocity, acceleration, force, momentum and so on.
As equations are written, show students how units for each quantity can be deduced from the equation. Have students determine the SI units before moving forward in the slide. This technique limits the amount of memorization required. See if students can suggest additional possible units of average velocity.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow them some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful. Show students how to obtain both answers to the first problem. For the second problem, point out the error in simply averaging the two velocities. This is wrong because the car spends more time traveling at the slower speed.
When discussing the second bullet point, ask students to describe the difference between distance and displacement. Then, ask students to explain why the third bullet point is true. (Answer: In a round trip, the displacement is zero, thus the average velocity is also zero. The speed is not zero because the distance traveled is not zero.)
Remind students that slopes have units. Many might just say that the slope is “1” instead of “1 m/s.”
Have students write their answers in their notes. Discuss the answer to object 1 before they answer questions 2 and 3. Many students will forget that velocity includes direction so they might simply answer “constant velocity” or “constant forward velocity”. This offers a chance to review the sign conventions for displacement and velocity.
When asking students to express their ideas, you might try one of the following methods. (1) You could ask them to write their answers in their notebook and then discuss them. (2) You could ask them to first write their ideas and then share them with a small group of 3 or 4 students. At that time you can have each group present their consensus idea. This can be facilitated with the use of whiteboards for the groups. The most important aspect of eliciting student’s ideas is the acceptance of all ideas as valid. Do not correct or judge them. You might want to ask questions to help clarify their answers. You do not want to discourage students from thinking about these questions and just waiting for the correct answer from the teacher. Thank them for sharing their ideas. Misconceptions are common and can be dealt with if they are first expressed in writing and orally. Students will often only choose the first option as an accelerating vehicle. They think of the others as decelerating and constant velocity. The car is accelerating in each example except for the cruise control. The first is positive acceleration, the second is negative acceleration, and the fourth is accelerating because direction is changing (and thus velocity is changing, even though speed is constant). Centripetal acceleration will be covered in a later chapter but it is good to introduce the idea here, so students realize that acceleration is any change in velocity (either a change in the magnitude of velocity, or a change in the direction of velocity, or both).
Have students analyze the equation before providing the answer to the units. Stress that m/s 2 are a short way of saying (m/s)/s. It is a good idea to keep saying (m/s)/s in order to emphasize the fact that acceleration is the change in velocity (m/s) over a period of time (s).
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow them some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful.
Equations (1) and (2) are the definitions of velocity and acceleration. Equations (3), (4), and (5) are only valid for uniform acceleration. Show students how to derive equation (4) by combining (1), (2), and (3). Then allow students some time to derive (5) from (1), (2), and (3) by eliminating time. Since (4) and (5) are derived from the first three, there are no problems that can be solved with them that could not have been solved by using the first three equations. It might be easier to use (4) and (5) but it is not necessary. They do not represent any “new” rules.
For problems, it is a good idea to go through the steps on the overhead projector or board so students can see the process instead of just seeing the solution. Allow them some time to work on problems and then show them the proper solutions. Do not rush through the solutions. Discuss the importance of units at every step. Problem solving is a developed skill and good examples are very helpful. After using equation (4) to solve the first problem, show the students that they would obtain the same answer by using equation (3) followed by equation (1). Similarly, the second problem can be solved with equation (5) or by using (3), then (1), then (2).
When presenting this slide, you may wish to refer students to Figure 14 in their textbook. You could also begin this slide with a video clip of the “feather and hammer” experiment on the moon. It is available on NASA and other web sites. Perform an internet search with the terms “feather and hammer on moon” to find a link. Discuss the effects of air resistance with students. With air resistance, an object will continue to accelerate at a smaller rate until the acceleration is zero. At that point the object has reached “terminal velocity.”
When presenting this slide, you may wish to refer students to Figure 15 in their textbook. You can also demonstrate the motion for students. Toss a ball up and catch it. Ask students to focus on the spot half-way up and observe the motion at that time. They can then predict the sign for the velocity and acceleration at that point. Then ask students to focus on the peak and, finally, on a point half-way down. Often students believe the acceleration at the top is zero because the velocity is zero. Point out to them that acceleration is not velocity, but changing velocity. At the top, the velocity is changing from + to -. Ask students to explain each combination above. For example, a positive velocity (moving upward) and a negative acceleration (downward) would cause the velocity to decrease.
Now students are asked to graph the motion they just observed. This graph should match the answers to the chart on the last slide. Remind them that they are graphing velocity, but acceleration is the slope of the velocity-time graph. Student graphs may have a different initial velocity and a different x -intercept (the time at which the velocity reaches zero), but their graphs should have the same shape and slope as the one given on the slide. Point out that the velocity is zero at the peak ( t = 1.1 s for this graph) while the acceleration is never zero because the slope is always negative. Help them get an approximate slope for the graph shown on the slide. It should be close to -9.81 (m/s)/s.
The equations from Section 2 apply because this is uniform acceleration. Simply use “ y ” instead of “ x, ” and the acceleration is -9.81 m/s 2 . Allow students some time to get the answers for t = 1.00 s, and then show them the calculations. Then have them continue with the following rows of the table. Students can use equation (4) from the previous lecture to get y , and the second version of equation (2) to get v . Or, they could get y by using equation (1) after getting the velocity, but they must get the average before using equation (1). Point out to students that the ball turns around between the 2.00 and 3.00 second mark. This makes sense, since it starts with a velocity of 15.2 m/s and loses 9.81 m/s of it’s velocity each second (in other words, the velocity decreases by 9.8 m/s in each step).
To make the situations more concrete, use an automobile as an example. For example, the first combination would be a car moving to the right (v is +) and accelerating to the right (a is +), so the speed will increase. Some students may be confused by the latter two examples, thinking that a negative acceleration corresponds to slowing down and a positive acceleration corresponds to speeding up. Emphasize that the directions of velocity and acceleration must both be taken into account. In the third example, the velocity and acceleration are in the same direction, so the object is speeding up. In the fourth case, they are in opposite directions, so the object is slowing down.
The equation for v avg is only valid if the velocity increases uniformly (a straight line in a velocity-time graph) or, in other words, if the acceleration is constant.
Students may think “B” is at rest and “C” is moving backwards or to the left. If so, they are confusing position-time graphs with velocity-time graphs. Ask them to look at “B” and think about what it means if velocity stays the same or look at “C” and ask them what it means if velocity is decreasing. A good exercise for the students at this time would be the use of the Phet web site: http://phet-web.colorado.edu/web-pages/index.html NOTE: These simulations are downloadable so you can avoid the need for internet access after a onetime download. If you choose the “Motion” simulations and then choose the “Moving man” option, the students can observe the motion of a man (constant velocity, speeding up, slowing down, at rest) and see the graphs of position-time, velocity-time and acceleration-time. You might start with “at rest” and ask them to predict the shape of each graph before running the simulation. Then ask them how each would change if he moved forward with a constant speed. Follow this with other changes, such as changing the starting position or accelerating the walker.
You will see me use d instead of x for displacement
6.
Displacement ( x )
Displacement is not always equal to the distance traveled
A gecko runs up a tree from the 20 cm marker to the 80 cm marker, then he retreats to the 50 cm marker?
7.
Displacement ( x )
In total it traveled 90 cm, however, the displacement is only 30 cm
x or d = 50cm- 20cm = 30 cm
If the gecko goes back to the start
The displacement is zero
8.
Displacement
What is the displacement for the objects shown?
Answer: 70 cm
Answer: -60 cm
9.
Displacement - Sign Conventions
Right (or east) ---> +
Left (or west) ---> –
Up (or north) ----> +
Down (or south) ---> –
10.
Average Velocity
The total displacement divided by the time interval during which the displacement occurred (V avg )
Basically velocity is distance/time
Average velocity is displacement divided by the time interval.
11.
Average Velocity
The units can be determined from the equation.
SI Units: m/s
Other Possible Units: mi/h, km/h, cm/year
Velocity can be positive or negative but time must always be positive
12.
Average Velocity
You travel 370 km west to a friends house. You left at 10am and arrived at 3pm. What was your average velocity?
V avg = d = -370 km
Δt 5.0h
V avg = -74 km/h or 74 km/h west
13.
Classroom Practice Problems
Joey rides his bike for 15 min with an average velocity of 12.5 km/h, how far did he ride?
V avg = d
Δt
d = (12.5 km/h) (0.25h)
d = 3.125 km
14.
Speed
Speed does not include direction while velocity does.
Speed uses distance rather than displacement.
In a round trip, the average velocity is zero but the average speed is not zero.
15.
Graphing Motion
How would you describe the motion shown by this graph?
Answer: Constant speed (straight line)
What is the slope of this line?
Answer: 1 m/s
What is the average velocity?
Answer: 1 m/s
16.
Graphing Motion
Describe the motion of each object.
Answers
Object 1: constant velocity to the right or upward
Object 2: constant velocity of zero (at rest)
Object 3: constant velocity to the left or downward
17.
2.2 Acceleration Pg. 48 to 59
18.
What do you think?
Which of the following cars is accelerating?
A car shortly after a stoplight turns green
A car approaching a red light
A car with the cruise control set at 80 km/h
A car turning a curve at a constant speed
Based on your answers, what is your definition of acceleration?
19.
Acceleration
What are the units?
SI Units: m/s 2
Other Units: (km/h)/s or (mi/h)/s
Acceleration = 0 implies a constant velocity (or rest)
20.
Classroom Practice Problem
A bus slows down with an average acceleration of -1.8m/s 2 . how long does it take the bus to slow from 9.0m/s to a complete stop?
Find the acceleration of an amusement park ride that falls from rest to a velocity of 28 m/s downward in 3.0 s.
9.3 m/s 2 downward
21.
Acceleration
Displacement with constant acceleration
d = ½ (v i + v f ) Δt
Displacement= ½ (initial velocity + final velocity) (time)
22.
Example
A racecar reaches a speed of 42m/s. it uses its parachute and breaks to stop 5.5s later. Find the distance that the car travels during breaking.
Given: v i = 42 m/s v f = 0 m/s
t = 5.5 sec d = ??
23.
Example
d = ½ (v i + v f ) Δt
d = ½ (42+0) (5.5)
d = 115.5 m
24.
Acceleration
Final velocity
Depends on initial velocity, acceleration and time
Velocity with constant acceleration
v f = v i + aΔt
Final velocity = initial velocity + (acceleration X time)
25.
Acceleration
Displacement with constant acceleration
d = v i Δt + ½ a (Δt) 2
Displacement = (initial velocity X time) + ½ acceleration X time 2
26.
Example
A plane starting at rest at one end of a runway undergoes uniform acceleration of 4.8 m/s 2 for 15s before takeoff. What is its speed at takeoff? How long must the runway before the plane to be able to take off?
Given: v i = 0 m/s a= 4.8m/s s
t= 15s v f = ?? d= ??
27.
Example
What do we need to figure out?
Speed at takeoff
How long the runway needs to be
v f = v i + aΔt
v f = 0m/s + (4.8 m/s 2 )(15 s)
v f = 72m/s
28.
Example
d = v i Δt + ½ a (Δt) 2
d = (0m/s)(15s) + ½ (4.8m/s 2 )(15s) 2
d = 540 m
29.
Acceleration
Final velocity after any displacement
v f 2 = v i 2 +2ad
Final velocity 2 = initial velocity 2 + 2(acceleration)(displacement)
30.
Example
An aircraft has a landing speed of 83.9 m/s. The landing area of an aircraft carrier is 195 m long. What is the minimum uniform acceleration required for safe landing?
Answer: -18.0 m/s 2
31.
Useful Equations
1.
2.
3.
4.
5.
d = 1 (v i + v f ) t 2
32.
Classroom Practice Problems
A bicyclist accelerates from 5.0 m/s to 16 m/s in 8.0 s. Assuming uniform acceleration, what distance does the bicyclist travel during this time interval?
Answer: 84 m
33.
2.3 Falling Objects
34.
Free Fall
The motion of a body when only the force due to gravity is acting
Acceleration is constant for the entire fall
Acceleration due to gravity ( a g or g )
Has a value of -9.81 m/s 2 (we will use 10m/s 2 )
Negative for downward
Roughly equivalent to -22 (mi/h)/s
35.
Free Fall
Acceleration is constant during upward and downward motion
Objects thrown into the air have a downward acceleration as soon as they are released.
36.
Free Fall
The instant the velocity of the ball is equal to 0 m/s is the instant the ball reaches the peak of its upward motion and is about to begin moving downward.
REMEMBER!!!
Although the velocity is 0m/s the acceleration is still equal to 10 m/s 2
37.
Example
Jason hits a volleyball so that it moves with an initial velocity of 6m/s straight upward. If the volleyball starts from 2.0m above the floor how long will it be in the air before it strikes the floor?
We want to use our velocity and acceleration equations
38.
Example
Given: v i = 6m/s a= 10m/s 2
d= -2.0m t = ?? v f = ??
v f 2 = v i 2 + 2ad
v f = at + v i
39.
Free Fall
For a ball tossed upward, make predictions for the sign of the velocity and acceleration to complete the chart.
Velocity (+, -, or zero) Acceleration (+, -, or zero) When halfway up When at the peak When halfway down + - zero - - -
40.
Graphing Free Fall
Based on your present understanding of free fall, sketch a velocity-time graph for a ball that is tossed upward (assuming no air resistance).
Is it a straight line?
If so, what is the slope?
Compare your predictions to the graph to the right.
41.
Classroom Practice Problem
A ball is thrown straight up into the air at an initial velocity of 25.0 m/s upward. Create a table showing the ball’s position, velocity and acceleration each second for the first 5 s.
20.1 +15.2 -9.81 30.4 +5.4 -9.81 30.9 -4.4 -9.81 21.6 -14.2 -9.81 2.50 -24.0 -9.81 t (s) y (m) v (m/s) a (m/s 2 ) 1.00 2.00 3.00 4.00 5.00
42.
Direction of Acceleration
Describe the motion of an object with
v i and a as shown to the left.
Moving right as it speeds up
Moving right as it slows down
Moving left as it speeds up
Moving left as it slows down
V i a + + + - - - - +
43.
Graphing Velocity
The slope (rise/run) of a velocity/time graph is the acceleration.
Rise is change in v
Run is change in t
This graph shows a constant acceleration.
Average speed is the midpoint.
44.
Graph of v vs. t for a train
Describe the motion at points A, B, and C.
Answers
A: accelerating (increasing velocity/slope) to the right
B: constant velocity to the right
C: negative acceleration (decreasing velocity/slope) and still moving to the right