Name:
IB Physics 1
Chapter 2 – One Dimensional Motion
Unit Objectives
Obj.
Assessment Statement
Notes
Distinguish between vector and scalar quantities, and
give examples of each.
A vector is represented in print by a bold italicized
symbol, for example, F.
Determine the sum or difference of two vectors by a
graphical method
Multiplication and division of vectors by scalars is also
required
Define displacement, velocity, speed and acceleration
Quantities should be identified as scalar or vector
quantities
Explain the difference between instantaneous and
average values of speed, velocity and acceleration.
Outline the conditions under which the equations for
uniformly accelerated motion may be applied
Identify the acceleration of a body falling in a vacuum
near the Earth’s surface with the acceleration g of free
fall
Solve problems involving the equations of uniformly
accelerated motion
Describe the effects of air resistance on falling objects
Draw and analyze distance–time graphs, displacement–
time graphs, velocity–time graphs and acceleration–time
graphs.
Calculate and interpret the gradients of displacement–
time graphs and velocity– time graphs, and the areas
under velocity–time graphs and acceleration–time
graphs.
Determine relative velocity in one and two dimensions
I.
Only qualitative descriptions are expected. Students
should understand what is meant by terminal speed.
Students should be able to sketch and label these graphs
for various situations. They should also be able to write
descriptions of the motions represented by such graphs.
Scalars and Vectors
A. Scalars are quantities that are represented by a number that provides the size (magnitude)
1. Temperature and mass are examples of scalars
2. When adding scalars, you follow typical addition rules (i.e. 4 + 3 = 7)
B. Vectors consist of both magnitude and direction
C. Vectors are represented as an arrow.
1. The direction of the arrow provides the direction of the vector.
2. The length of the vector arrow is proportional to the magnitude of the vector.
D. Vector addition is dependent upon direction of arrows.
1. When adding vectors, the first step is to make sure they are positioned head to tail
2. The sum, called the resultant, is the vector that runs from the tail of the initial vector to the
head of the last vector.
II. Distance vs. Displacement
A. Kinematics is the study of motion. (Does not explain why an object moves, just how)
B. Distance, a scalar, is length of the path traveled.
C. Displacement is a vector and is defined as the length from an object’s initial position to its final
position.
D. x = x – xo
where x is the final position and xo is the initial position
III. Speed and Velocity
A. Speed is a scalar quantity that gives the rate at which distance is covered by a moving object.
distance
1. speed =
time
2. Speeds are always positive
B. Velocity is a vector that provides the rate in change of position over a time interval
∆𝑥
1.
v=
2.
Velocities can be negative if displacement is negative.
∆𝑡
C. Instantaneous velocities are velocities at a specific instant in time (Δt will be a small interval).
Average velocities are velocities over a larger time interval.
D. Instantaneous and average velocity values are the same if velocity remains constant (no acceleration)
Sample Problem #1
In the 2000 Olympic Games, Michael Johnson won the 400 m race (once
around the track) in 43.49 s. Find his average velocity. Find his average
speed.
Sample Problem #2
The three toed sloth, the slowest mammal known, travels a blazing 15
yards/minute. Determine how long it would take a sloth to cover 1 m. (Hint: 1
m = 3.281 ft)
Sample Problem #3
A pickup truck travels 5.2 miles at 43 mph. The truck runs out of gas, so the driver walks 1.2 miles. It takes the driver 27 min to
walk that distance. What is the average velocity for the entire 6.4 mile trip?
IV. Acceleration
A. Acceleration is a vector that represents the change in an object’s velocity
∆𝑣
B. a =
∆𝑡
C. This equation shows the change in velocity every second. Therefore, the unit for acceleration is m/s/s
or m/s2 (properly written as m s-2)
D. A positive acceleration means the object is increasing its velocity. A negative acceleration shows the
object is decreasing its velocity.
Velocity
Acceleration
Physical Meaning
+
+
speeding up in forward direction
+
slowing down while moving backwards
+
slowing down while moving forwards
speeding up in backward direction
E. Summary
1.
2.
3.
An object will speed up if and only if v and a point in the same direction
An object will slow down if and only if v and a point in opposite directions
An object moves at constant velocity only if a = 0.
Sample Problem #4
A drag racer crosses the finish line and deploys the parachute. At t = 9.0s, the car’s velocity is 28 m s-1. At 12.0 s, the velocity is
reduced to 13 m s-1. What is the average acceleration of the car over this time interval?
Conceptual Question #1
a. Is it possible to have a large velocity and no acceleration?
b. Is it possible to have an acceleration and little velocity?
c. Is it possible to have an acceleration and no velocity?
d. Is it possible to have the acceleration and velocity vectors point in opposite directions?
e. Is it possible to accelerate and not change speeds?
V. Graphical Analysis – Displacement Time Graphs
A. From section III we have average velocity v =
B.
C.
D.
E.
F.
G.
H.
∆𝑥
∆𝑡
y
The slope of any line is as m =
x
If we plot x (displacement) on the y axis and t on the x axis, the slope will provide the velocity.
The steeper the slope, the faster the object is moving.
If the slope is 0, the object is at rest.
The object is moving at a constant velocity if and only if the slope is constant (the plotted function
must be a line)
Accelerated motion will produced a curved function
To find the instantaneous velocity, find the slope at a specific instant. There are several ways to do
this.
1. You can find the approximate slope over a very small time interval.
2. You can draw a tangent line to the curve and find its slope
3. Another method is to use calculus to find the derivative
VI. Graphical Analysis – Velocity Time Graphs
A. Using the same reasoning as covered in section V, the slope of a v vs. t plot will provide the
acceleration
B. The steeper the slope, the larger the acceleration, meaning the faster the velocity changes.
C. A slope of 0 means the object is not accelerating. Therefore, the object maintains constant velocity.
D. The area under the curve of a velocity time graph represents displacement.
VII. Constant Acceleration
A. All problem solving that we will attempt will involve constant acceleration
B. All equations for accelerated motion are derived from the definition of acceleration.
v = vo + at
x = ½at2 + vot
v2 = vo2 + 2ax
x = ½(vo + v)t
Problem Solving Steps
1.
2.
3.
4.
5.
6.
7.
Read the problem carefully!
Sketch the situation. Label all vectors.
Identify what variables you know.
Decide what variable(s) you are trying to determine
Based on what is known and what you are trying to find, choose the appropriate equation(s). Sometimes you will
need to use more than one equation to get the correct answer!!!
Plug and chug. Repeat if needed.
Check units on final answer and check “Is the answer reasonable?”
Sample Problem #6
A car enters the on ramp of a highway at an initial velocity of 22 m s-1 and begins to accelerate at 3.2 m s-2. Determine how far
the car has traveled after 5.4 s.
Sample Problem #7
A spaceship is traveling at 3250 m s-1. As its retrorockets are fired, the ship begins to slow down with a magnitude of 10.0 m s-2.
Determine the velocity of the ship after a displacement of 215 km relative to the point where the rockets were fired.
VIII. Free Fall Acceleration (2.6; 2.1.8, 2.1.9, 2.1.10)
A. Free fall is a special situation in which the only force that acts upon an object is
the force of gravity. By definition, there can be no other forces on the object,
including air resistance when an object is in free fall.
B. If there is no air resistance, all objects, regardless of mass, are pulled downward
with the same acceleration.
C. If the distance of the fall is small compared to the radius of the earth, the
acceleration is constant
D. The acceleration due to gravity, g, is assumed to have a constant value, 9.8 m/s2
or 32 ft/s2, towards the center of the earth.
E. In free fall the acceleration is assumed to be constant so we will use equations of
accelerated motion (see equations in section V).
F. Sometimes you will see minor changes to the accelerated motion equations.
1. Typically vertical motion is identified as y so x becomes y
2. Because the acceleration is the special case of free fall acceleration a becomes g
Sample Problem #8
In 1939, Joe Sprinz tried to set a world record for catching the highest dropped ball. A blimp hovering at 270 m (800 ft) above
Joe dropped the ball. If air resistance is ignored, determine the time it takes the ball to reach Joe.
Determine the velocity of the ball right before it hits Joe’s glove.
IX. Objects Projected Vertically
A. Objects that are thrown upward and return to earth exhibit symmetry in motion
B. The time up to the peak of flight is equal to the time back down to the starting point
C. The velocity at the peak is 0 m/s
D. Velocities up and down have the same magnitude provided they are at the same position.
E. Acceleration remains constant throughout and is equal to the acceleration due to gravity. Note that
the acceleration is equal to g and always equal to g once the object is in the air. The acceleration
does not equal g when the object is in contact with the launcher.
Sample Problem #9
A person standing on a cliff 1.0 x 101 m above the ground throws a ball upward with an initial velocity of 12 m/s. Determine
how long after being thrown upward will the ball hit the ground 1.0 x 10 1 m below.
X. Acceleration with Air Resistance
A. Free fall acceleration is an idealized case as all objects outside a vacuum will
experience air resistance.
B. The air resistance will have a damping effect on the acceleration due to
gravity causing the actual acceleration during fall to be less than g.
C. As the object falls faster and faster, the force of air resistance increases which
causes the acceleration to become smaller.
D. Eventually, the acceleration will be 0 and the object will fall at its terminal
velocity.
E. Terminal velocity is a constant velocity because the acceleration is 0.
Sample Multiple Choice Questions
1. A motor car travels on a circular track of radius a, as shown in the figure. When the car has traveled from
P to Q its displacement from P is
A. a 2 SW
B. a 2 NE
C. 3a/2 SW
D. 3a/2 NE
N
P
W
a
Q
2.
A ball is released from rest near the surface of the Moon. Which one of the following quantities increases at a constant
rate?
A. Only distance fallen
C. Only speed
B. Only speed and distance fallen
D. Only speed and acceleration
3.
Peter and Susan both stand on the edge of a vertical cliff. Susan throws a stone vertically downwards and, at the same time,
Peter throws a stone vertically upwards. The speed V with which both stones are thrown is the same. Neglecting air
resistance, which one of the following statements is true?
A. The stone thrown by Susan will hit the sea with a greater speed than the stone thrown by Peter.
B. Both stones will hit the sea with the same speed no matter what the height of the cliff.
C. In order to determine which stone hits the sea first, the height of the cliff must be known.
D. In order to determine which stone hits the sea first both the height of the cliff and the mass of each stone must be
known.
E
S
4.
The graph shows the variation with time t of the velocity v of an object.
v
t
Which one of the following graphs best represents the variation with time t of the acceleration a of the
A. a
B. a
object?
0
C.
0
t
a
0
5.
0
D.
0
t
0
t
0
t
a
0
A car accelerates uniformly from rest. It then continues at constant speed before the brakes are applied,
bringing the car to rest. Which of the following graphs best shows the variation with time t of the
acceleration a of the car?
A.
a
0
B.
0
t
0
t
0
t
a
0
D.
t
a
0
C.
0
a
0
6.
The diagram below shows the variation with time t of the velocity v of an object.
v
0
0
t
The area between the line of the graph and the time-axis represents
A.
the average velocity of the object.
C. the acceleration of the object
B.
the displacement of the object.
D. the instantaneous velocity of the object
Textbook Questions/Problems
Conceptual Questions
1. One of the following statements is incorrect. (A) The car travelled around the oval track at a constant
velocity. (B) The car travelled around the oval track at a constant speed. Which statement is incorrect and
why?
2. An object moving with a constant acceleration can slow down. Can the object come to a complete stop if
the acceleration is constant? Explain.
3. Two balls are thrown straight up in the air from the same launch point. One ball is thrown before the
other. Is it possible that both balls will reach the same maximum height at the same instant>
Problems
1. A 16 year old runner can run a 10.0 km course with an average speed of 4.38 m/s. A 50 year older runner
runs the same course with an average speed of 4.27 m/s. If both runners start at the same time, how long
does the 16 year old wait for the 50 year old to cross the finish line?
2. A tourist is being chased by an angry bear. The tourist runs in a straight line towards his car at a speed of
4.0 m/s. The bear is 26 m behind the man and is chasing him with a speed of 6.0 m/s. How far can the man
run before the bear catches him?
3. A sprinter explodes out of the starting blocks with an acceleration of 2.3 m/s 2. She accelerates for 2.0 s
before she runs with a constant speed for the rest of the race. A) What is her speed after 2.0 seconds and b)
what is her speed at the end of the race?
4. A truck traveling with a speed of 33 m/s comes to a halt by decelerating at a rate of -11 m/s2. How far
does the truck travel while in the process of stopping?
5. A car is traveling at a constant speed of 12 m/s. The driver sees a red light. After 0.210 s have elapsed
after seeing the light, the driver hits the brakes and the car decelerates at a rate of -5.4 m/s2. How far does
the car travel while stopping, as measured from the point where the driver first sees the light turn red?
6. A girl drops a water balloon from a bedroom window. The window is 6.0 m above the ground. How long
does it take before the balloon hits the ground?
7. A rope is being used to vertically lift a piano into an apartment. The rope breaks and the the piano
accidentally falls from rest from a height of 12.0 m. When the piano is 7.5 m above the ground, a 2.0 m
tall man notices the piano is directly above him. How much time does the man have to get out of the way?
8.
Using the x-t graph below, calculate the average velocity in km/hr for segments A, B, and C.
Mark any times on the graph where the object has accelerated.
9.
Using the v-t graph below, calculate the average acceleration during segments A, B and C.
10. Using the v-t graph below, calculate the displacement travelled by the object.
Is the displacement greater than, less than, or equal to the distance travelled by this object? How can you
tell?
Section 2.6
Section 2.7
41, 43 - 3.06 s, 45, 51, 53 – 0.405 s , 55 – 0.93 m/s
62, 63