1
Week 3
SELF-LEARNING PACKAGE IN
GENERAL PHYSICS 1
Uniformly Accelerated Motion
Learning Competency:
•
Convert a verbal description of a physical situation involving uniform acceleration in one dimension
into a mathematical description STEM_GP12Kin-Ib-12
•
Interpret displacement and velocity, respectively, as areas under velocity vs. time and acceleration vs.
time curves STEM_GP12KINIb-14
•
Interpret velocity and acceleration, respectively, as slopes of position vs. time and velocity vs. time
curves STEM_GP12KINIb-15
•
Construct velocity vs. time and acceleration vs. time graphs, respectively, corresponding to a given
position vs. time-graph and velocity vs. time graph and vice versa STEM_GP12KINIb-16
•
Solve for unknown quantities in equations involving one-dimensional uniformly accelerated motion ,
including free fall motion STEM_GP12KINIb-17
•
Solve problems involving one-dimensional motion with constant acceleration in contexts such as, but
not limited to, the “tail-gating phenomenon”, pursuit, rocket launch, and freefall problems
STEM_GP12KINIb-19
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Ready to Launch!
If you throw a ball up, how high will it be? What can you say about the
acceleration of a basketball thrown to the basket? Most often, acceleration takes
place in every moving object, and they may be uniform or non-uniform. Whatever the case, motion is observe everywhere. As they say, no particle in the universe is absolutely at rest.
Our exploration of the physical world will start at the concept of Mechanics, that is, observing relationships between matter, forces, and motion. No
doubt, these three are basic concepts of Physics. Specifically, the concept of motion involves many physical quantities which maybe familiar to us. These are
velocity, speed, distance, time, displacement and acceleration. This week, we
will start with the simplest of motion– that is motion along a straight path.
Aim at the Target!
For this topic, you explore the following objectives:
1. Describe a physical situation involving uniform acceleration in one dimension
verbally and mathematically
2. Interpret displacement and velocity, respectively, as areas under velocity vs.
time and acceleration vs. time curves,
3. Interpret velocity and acceleration, respectively, as slopes of position vs. time
and velocity vs. time curves ,
4. Construct velocity vs. time and acceleration vs. time graphs, respectively, corresponding to a given position vs. time-graph and velocity vs. time graph and
vice versa,
5. Solve for unknown quantities in equations involving one-dimensional uniformly accelerated motion , including free fall motion, and
6. Solve problems involving one-dimensional motion with constant acceleration.
Try This!
•
List the things that you find common to a
sliding ball and that of the falling apple
•
How about if you change the object in the
illustration, instead of a ball, it is a wooden
block, and instead of falling apple, it is a falling moon; Do you think your list will
change?
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Keep This in Mind!
ACTIVITY (DAY 1)
Task 1A: Increasing or Decreasing?
Instruction:
1. Position a long inclined plane( with height at least 50 cm) as shown
in the figure below.
2. From the top part of the plane, measure three intervals 30cm apart
up to the bottom part.
30cm
30cm
30cm
3. Put a blockage (e.g ruler or book} on the first interval, then position
the ball or any round object at the top end of the inclined plane. Using a stopwatch or timer in a cellphone, time the duration of the ball as it rolls from the
topmost end of the plane up to the time it hits the blockage. Record the time
elapse. Do this procedure two more times.
30cm
30cm
TIME IT!
30cm
4. Repeat procedure #3 but put the blockage to the second interval for
three trials, then on the third interval for another three trials.
30c
30c
30c
30c
5. Complete the table on the next page.
30c
30c
4
Table 1. Table for Task 1A
Category
Trial 1
(Time)
Speed
Trial 2
(Time)
Speed
Trial 3
(Time)
Speed
First Block
Second
Third
Note: d1 = 30 cm
d2 = 60cm
d3 = 90cm
Task 1B: Increasing or Decreasing?
Instruction:
1. Set up same material as that in task 1A but this time with base blanket or
any plane cloth on the floor.
2. , Measure three positions 30cm apart where the topmost part of the plane to
be the first and the third somewhere in the middle part (refer to the illustration below).
30cm
30cm
30cm
Blanket
3. Put the ball on the first position( at the topmost part), then let it roll down up
to the bottom part. Measure the time duration for the ball to reach the bottom part of
the plane. The time( to )measured here will be done only once.
4. Put again the ball on the first position( at the topmost part), then let it roll
down up to the bottom part and continue along the blanket until it stop. Measure the
time duration from release up to the time it will stop. Measure also the distance covered by the ball from the base of the inclined plane up to the point it stop. Do this in
three trials. Record the data.
6. Repeat procedure #4, this time the ball to be put on second and third position.
7. Complete the table on the next page.
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Table 2. Table for task 1B.
Position
to
t1
d1
t2
d2
t3
d3
1st
2nd
3rd
Note: *to solve for t1, t2, t3; t1 or t2 or t3 = t in procedure #4 minus to in
procedure #3. Also, to is measured from the initial point(1st, 2nd, and 3rd position) to the base of incline plane.
ANALYSIS(DAY 1)
To deepen your ideas about the activities, you may answer in a separate
physics notebook all guide questions provided;
1. Based from the activity, how will you verbally define acceleration? Did
the activity showed uniformly accelerated motion? Explain.
2. Mathematically, how will you describe acceleration? How about uniformly accelerated motion?
3. There are factors to consider for a motion to have uniform acceleration, in the activity, what are some of the considerations you think should be
present so that what you observe is really uniformly accelerated motion?
4. What causes the ball to accelerate in the first part of the activity and
for it to decelerate in the second part? Explain.
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ACTIVITY (DAY 1)
Task 2A: Amazing Bound Areas
Instruction:
Velocity (m/s, East)
Given a graph of velocity vs time, calculate for the bound (shaded)
area represented by different color using the area of triangle and/or the area
of rectangle, then answer the following questions
3
2
B
C
1
4
D
A
Time (second)
Area A = _____________________
Area B = _____________________
Area C = _____________________
Area D = _____________________
•
Which graph/s has constant velocity? _______________
•
Which graph/s the object moves due west?_____________
•
Which graph/s shows object speeds up?_____________
•
Which graph/s the object proceeds the longest displacement?____________
•
Which graph/s the object slow down?______________
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ACTIVITY (DAY 1)
Task 2B: Amazing Bound Areas
Instruction:
Acceleration (m/s2), East)
Given a graph of acceleration vs time, calculate for the bound (shaded)
area represented by different color using the area of triangle and/or the area
of rectangle, then answer the following questions below.
2
3
1
B
C
4
A
D
Time (second)
Area A = _____________________
Area C = _____________________
Area B = _____________________
Area D = _____________________
•
Which graph/s has constant acceleration? __________
•
Which graph/s the object moves due east?_________
•
Which graph/s has endpoint showing approaches constant velocity?______
•
Which graph/s the object proceeds the highest velocity?_________
•
Which graph/s shows the object changes acceleration?_________
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ACTIVITY (DAY 2)
Task 3A: Look at the Slope!
Instruction:
Displacement (m, North)
Given a graph of displacement vs time, calculate for the slope of each
line segments, then answer the following questions below.
C
B
D
A
Time (second)
Slope A = _____________________
Slope C = _____________________
Slope B = _____________________
Slope D = _____________________
•
Which graph/s has constant velocity? _______________
•
Which graph/s the object is not moving?_____________
•
Which graph/s the object moves in opposite direction?_____________
•
Which graph/s the object proceeds the highest velocity?______________
•
Which graph/s shows the object accelerates?______________
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ACTIVITY (DAY 2)
Task 3B: Look at the Slope!
Instruction:
Given a graph of velocity vs time, calculate for the slope of each line segments, then answer the following questions below.
Velocity (m/s, North)
E
C
D
B
A
Time (second)
Slope A = ___________
Slope B = ___________
Slope C = ___________
Slope E = ___________
Slope D = ___________
•
Which graph/s has constant acceleration? _______________
•
Which graph/s the object has highest magnitude for its acceleration?______
•
Which graph/s the object moves in opposite direction?_____________
•
Which graph/s the object is at rest?______________
•
Which graph/s shows the object moves with constant velocity?__________
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ACTIVITY (DAY 2)
Task 4A: Morphing graphs!
Instruction:
Displacement (m, North)
Given a graph of displacement vs time, complete the table below using
the slope formula, v= (d2-d1/t2-t1), and then construct a corresponding velocity vs time graph using the data.
D
C
B
A
A
Time (second)
1
Time (second)
2
3
4
5
6
7
Use the data in the
table to construct a
velocity vs time
graph, you may use a
colored pen / ballpen. Utilize an empty
plane on the right.
velocity (m/s, North)
Velocity (m/s)
Time (second)
8
9
10
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ACTIVITY (DAY 2)
Task 4B: Morphing graphs!
Instruction:
Given a graph of velocity vs time, complete the table below by finding the
bound area correspond to a line segment, d= bound area, and then construct
a corresponding displacement vs time graph using the data.
velocity (m/s, North)
D
C
B
A
A
Time (second)
1
Time (second)
2
3
4
5
6
7
Use the data in the
table to construct a
displacement vs
time graph, you
may use a colored
pen / ballpen. Utilize an empty cartesian plane on the
right.
displacement( m, North)
Velocity (m/s)
Time (second)
8
9
10
12
ACTIVITY (DAY 2)
Task 4C: Morphing graphs!
Instruction:
Velocity (m/s, North)
Given a graph of velocity vs time, complete the table below using the
slope formula, a= (v2-v1/t2-t1), and then construct a corresponding acceleration vs time graph using the data.
C
A
A
Time (second)
D
B
1
Time (second)
2
3
4
5
6
7
Use the data in
the table to construct a acceleration vs time graph,
you may use a colored pen / ballpen. Utilize an
empty plane on
the right.
Acceleration (m/s2, North)
Acceleration
Time (second)
8
9
10
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ACTIVITY (DAY 2)
Task 4D: Morphing graphs!
Instruction:
Acceleration (m/s2, East)
Given a graph of acceleration vs time, complete the table below by finding the bound area correspond to a line segment, v= bound area, and then
construct a corresponding velocity vs time graph using the data.
D
B
C
A
Time (second)
Time (second)
1
2
3
4
5
6
7
Use the data in the
table to construct a
velocity vs time
graph, you may
use a colored pen /
ballpen. Utilize an
empty cartesian
plane on the right.
velocity( m, East)
Velocity (m/s)
Time (second)
8
9
10
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ANALYSIS(DAY 1 and 2)
To deepen your ideas about the activities, you may answer in a separate
physics notebook all guide questions provided;
1. What does the bound area represent in velocity time graph?
2. What does the bound area represent in acceleration vs time graph?
3. In a velocity vs. time graph, what does a slant straight line means?
4. Recalling your topic in Basic calculus, how is this activity related with
definite integral?
5. Looking at the graphs, when can you say that a moving body decelerate (has negative acceleration)?
6. Can we use bound area to solve for the acceleration? Explain.
7. How do finding bound area and finding slope of the line differs in
terms of the quantity they represent?
8. What does the slope represent in displacement vs time graph? In velocity vs. time graph?
Self—Test
Recalling your topics in slope of
tangent line and definite integral,
find the following:
velocity
A. m(x) of the graph y =x2
B.
1. Which quantity associated to motion does m(x) is
equivalent?
2. Which quantity associated to motion does definite
integral of the graph is
equivalent?
time
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ACTIVITY (DAY 3)
Task 5A. Let the unknown be Revealed!
Instruction:
1. From the previous activities, you have encountered quantities
such as displacement (position), speed and velocity, and acceleration. This time around, given are equations/formulas for these
quantities, you are task to solve for the unknown quantities ask in
the problem.
2. Supply a data for each box provided for each problem in this activity.
First equation:
2nd Equation:
a = constant acceleration
a = constant acceleration
vf = final velocity
vf = final velocity
vi = initial velocity
vi = initial velocity
= Net Displacement
= time elapsed
3rd Equation:
a = constant acceleration
vi = initial velocity
= time elapsed
Continue to the next page…
= Net Displacement
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Carefully analyze the following problems and solve for the unknown quantities.
1. The car moves on a very long straight highway at an initial velocity of 2.0
m/s when it started to accelerate at a rate of 1.5 m/s 2. When the car
reaches a displacement of 1km, it started to decelerate at constant rate
until it stop.
A. What is the car’s final velocity( assumed that the car directs in a constant
direction) before it started to decelerate (slow down)?
Write here the known quantities (Given):
Write here the unknown quantity
(Required):
Write here the most
appropriate formula:
Write here your step by step solution:
B. What it the car ‘s deceleration?
Write here the known quantities (Given):
Write here the unknown quantity
(Required):
Write here your step by step solution:
Write here the most
appropriate formula:
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2. A ball assumed to be on level ground is exerted by an upward initial speed
of 25 m/s. Note that air resistance is negligible and acceleration due to gravity is equal to –9.8 m/s2;
A. How high will the ball go before it will start to fall down?
Write here the known quantities (Given):
Write here the unknown quantity
(Required):
Write here the most
appropriate formula:
Write here your step by step solution:
B. How much time the ball requires to reach back to level ground?
Write here the known quantities (Given):
Write here the unknown quantity
(Required):
Write here the most
appropriate formula:
Write here your step by step solution:
FYI: It is not surprising that free-falling motion when air resistance is negligible, indeed is also a uniformly accelerated motion.
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ABSTRACTION AND GENERALIZATION
In a real world, we can seldom
observe this what we call uniformly
accelerated motion (UAM) or motion
governed by a constant acceleration,
for some obvious reasons such as the
existence of variable frictions and other external forces, and that variability
of directions in the course of motion.
For the sake of this discussion, friction
maybe assumed negligible, and for
simplicity, the objects are assumed to
be moving in one dimension (straight
path).
1.1 Definition
of Uniformly Accelerated Motion (UAM)
By description, a moving body or object accelerate when;
•
It changes its speed ( speed up or slow down)
•
It changes direction ( force changes direction)
•
It changes both its speed and direction
Verbally, Uniformly Accelerated Motion (UAM) means that the object
changes its speed uniformly or in constant rate in only one direction. In
the case of free falling motion, acceleration always point towards earth’s
center (downward).
On one hand, mathematically UAM can be defined as uniform rate of
change in velocity, or the ratio between velocity and time elapsed remains
the same although out the course of motion toward a constant direction.
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2.1 Bound Area of a Velocity vs Time Graph
•
Graph 1 shows Uniformly Accelerated
Motion (uniform speed
increase)
•
Graph 2 shows Uniformly Decelerated
Motion (uniform speed
decrease)
Velocity (m/s,
2
C
1
B
A
Time (second)
The Bound Area of a velocity vs time graph represents the displacement made
by the moving object for a certain time duration. Mathematically, this can be solve
using the Riemann Sum of simply a definite integral. In the illustration above, solving for bound area will only utilize area of rectangle and/or triangle.
To solve for displacement made by
an object from 0 to 3 seconds;
To solve for displacement made by an
object from 3 to 7 seconds;
Displacement = Area A
Displacement = Area B +Area C
Area A is an area of triangle with base
3units and height 8 units
Area B is an area of rectangle with base
4units and height 5 units, while Area C
is an area of triangle with base 4 units
and height 3 units.
A = 0.5(base)(height) , base = 3 , height = 8
D1 = A = 0.5 (3 sec)(8m/s)
D1= 0.5(24) = 12 m, East
Area B = (base)(height) , base = 4
height = 5
= (4 sec)(5m/s) = 20 m
Area C = 0.5(base)(height), base = 4 height = 3
= 0.5 (4 sec)(3 m/s)
Area C = 0.5(12) = 6 m
So, displacement made from 0 to 7 seconds is D= D1 +D2
= 12 m + 26 m= 38 m, East
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Acceleration (m/s2, East)
2.1 Bound Area of a Acceleration vs Time Graph
•
Graph 1 shows
Non-uniform Accelerated Motion
•
Graph 2 shows
Non-uniform Decelerated Motion
2
C
1
B
A
Time (second)
The Bound Area of acceleration vs time graph represents the change in
velocity (instantaneous velocity) by the moving object for a certain time duration. Mathematically, this can be solve using the Riemann Sum of simply a
definite integral. In the illustration above, solving for bound area will only
utilize area of rectangle and/or triangle.
To solve for velocity by an object at
the end of 3 seconds;
To solve for change in velocity by an
object from 3 to 7 seconds;
v1 = Area A
Displacement = Area B +Area C
Area A is an area of triangle with base
3units and height 8 units
Area B is an area of rectangle with base
4units and height 5 units, while Area C
is an area of triangle with base 4 units
and height 3 units.
A = 0.5(base)(height) , base = 3 , height = 8
D1 = A = 0.5 (3sec)(8m/s2)
D1= 0.5(24) = 12 m/s, East
Area B = (base)(height) , base = 4
height = 5
= (4 sec)(5m/s2) = 20 m/s, East
Area C = 0.5(base)(height), base = 4 height = 3
= 0.5 (4 sec)(3m/s2)
Area C = 0.5(12) = 6 m/s, East
So, change in velocity from 0 to 7 seconds is v= v1 + v2
= 12 m/s+ 26 m/s= 38 m/s, East
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Displacement (m, North)
3.1 Slope of a Given graph
B
A
A
Time (second)
Definition of Velocity
velocity (m/s, North)
It is defined as the rate of change in displacement. In short, the slope of
a line segment in a displacement vs. time graph represent the change in velocity for a specific time duration.
B
A
A
Time (second)
Definition of Acceleration
Acceleration is defined as the rate of change in velocity. Likewise, the
slope of a line segment in a velocity vs. time graph represent the acceleration
for a specific time duration.
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Basic Equations Used for Uniformly Accelerated Motion( UAM)
First equation:
2nd Equation:
a = constant acceleration
a = constant acceleration
vf = final velocity
vf = final velocity
vi = initial velocity
= time elapsed
vi = initial velocity
= Net Displacement
3rd Equation:
4th Equation:
a = constant acceleration
vi = initial velocity
= time elapsed
= Net Displacement
= average velocity
vf = final velocity
vi = initial velocity
Note: All equations presented above are TRUE only for Motion with constant acceleration
General Equations Used for Uniformly Accelerated Motion( UAM)
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Reflect
Motion is everywhere, and it happens every time. As they say it is a
change in position with respect to a reference. Though, seldom we can observe uniformly accelerated motion, it is our first step to visualize our physical world. In your physics notebook, in reflection section, kindly comply the
tasks given.
Please Do it!
 What are some of the reasons why UAM is rarely observe in real environ-
ment?
 When the moving object changes direction just like object moving in a circular path, is it accelerating? If yes, is it uniformly accelerated motion? Explain.
 Make concept map / mind map about uniformly accelerated Motion.
Reinforcement & Enrichment
Practice Exercises
A. Make a velocity vs time graph of the following succession of events.
1. From 0 to 4 seconds, the object constantly accelerate at 2 m/s2
2. From 4 to 8 seconds, the object constantly decelerate at -1 m/s2
3. From 8 to 10 seconds, the object moves with constant velocity
4. From 10 to 12, the object constantly decelerate at until it stop
B. Make a displacement vs time graph of the following succession of events.
1. From 0 to 5 seconds, the object moves with constant velocity at 2 m/s
2. From 5 to 8 seconds, the object constantly decelerate at -1 m/s2
(the portion of this graph is parabolic)
3. From 8 to 12 seconds, the object remain at rest
C. Solve the following:
1. A motorcycle moves with constant acceleration with magnitude
3.0 m/s2 from location A to B. If it reaches to B in 1 minute,
(A) how far is B from A?
(B) how fast is the motorcycle upon reaching B if it started at rest in A?
(C) if location C is 2 km from B, and the motorcycle starts to decelerate at
constant rate of –1 m/s2 the moment it passes point B, can it reaches
C? if yes what it is its speed upon reaching C?
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Assess Your Learning
Velocity (km/hr, North)
Mr Obal jogged going to his grandfather in Barangay Ogtong due north
of his house. His travel was described by the graph as shown below:
B
C
A
D
A
Time (hour)
Based on the given graph, answer the following questions:
1. At what time after he left his house did he travel the fastest?
A. 2 hours
B. 5 hours
C. 7 hours
D. 10 hours
2. At which segment/s of his travel did he accelerate ?
A. A only
B. B only
C. B and D
D. A , C, and D
3. At what segment did he accelerated with highest magnitude?
A. A
B. B
C. C
D. D
References & Photo Credits
1. Sears & Zemansky (2008). University Physics with modern Physics.(12th
Edition). Pearson Addison-Wesly Publishing. Pp. 47-59.