PHYSICS 71
TOPICS
KINEMATICS IN ONE DIMENSION
Review on Vectors
Distance and Displacement
Speed, average, and
instantaneous velocity
Average and instantaneous
acceleration
Special Cases
MOTION IN 2 AND 3 DIMENSIONS
Projectile motion
Circular motion
Relative velocity
A= 4i +3j; x and y
Conversion of Magnitude direction to
Component - unit vector style and vice
versa
Component
With respect to x-axis
𝐴𝑥 = 𝐴 𝑐𝑜𝑠θ
𝐴𝑦 = 𝐴𝑠𝑖𝑛θ
With respect to y-axis
𝐴𝑥 = 𝐴𝑠𝑖𝑛θ
𝐴𝑦 = 𝐴 𝑐𝑜𝑠θ
Direction
𝐴=
2
2
𝐴𝑥 + 𝐴𝑦
( )
−1 𝐴𝑦
𝐴𝑥
θ = 𝑡𝑎𝑛
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MODULE 0: Review on Vectors
A. Scalar vs. Vector
Scalar: Quantity in magnitude only
Vectors: Quantity with both magnitude and
direction
Symbols: Bold letters, Letter with arrow
above, arrow lines, Magnitude of a vector
Reference frame:
Rectangular coordinate system:
azimuthal; considers whole coordinate
system as a circle (360deg)
Geographic coordinate system: North,
East, West, and South
Notations:
Magnitude direction:
A= 5, 36.87 deg
Component-Unit vector style:
B. Operation
a. Vector Addition /
Subtraction
𝑅= 𝐴±𝐵
𝑅 = (𝐴𝑥 ± 𝐵𝑥)𝑖 ± (𝐴𝑦 ± 𝐵𝑦)𝑗 ± (𝐴𝑍 ± 𝐵𝑍)𝑘
b. Scalar Multiplication/Dot
Product
𝑅 = 𝐴𝑥𝐵𝑋 + 𝐴𝑦𝐵𝑦 + 𝐴𝑧𝐵𝑧
Instantaneous Speed/Velocity: Velocity or
speed at a given instant
∆𝑥
𝑑𝑥
𝑣𝑖𝑛𝑠𝑡 = lim ∆𝑡 = 𝑑𝑡 ; derivative of
∆𝑡 → 0
displacement
c. Vector Multiplication /
Cross Product
𝑅 = 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝐴 × 𝐵
Magnitude: A x component of B that is
perpendicular to A
Direction: Perpendicular to both A and B
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Module 1: Kinematics in 1 dimension
Kinematics: Description of motion
All bodies are represented as a point
particle to described the motion of a
body/object
Point particle: representation of a material
which internal structure is ignored
Rectilinear motion: all motion are in a
straight line
A. Distance vs Displacement
Distance: total length; scalar
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒: 𝐴 + 𝐵 = 𝐶
Displacement: Shortest distance; Vector
𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡: 𝐴 − 𝐵 = 𝐶, 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
B. Speed vs Velocity vs
Instantaneous speed/velocity
Speed: change of distance with respect to
time; scalar
𝑆𝑎𝑣𝑒 =
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑖𝑚𝑒
Velocity: change of displacement with
respect to time; vector
𝑉𝑎𝑣𝑒 =
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑡𝑖𝑚𝑒
C. Acceleration
Change of velocity with respect to time
Can change magnitude and or direction
Direction of acceleration: increasing
velocity or decreasing velocity
-
Acceleration can happen if the
direction of an object changes or if
the velocity of an object decreases
Average acceleration: 𝑎𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =
𝑣𝑓−𝑣𝑖
∆𝑡
Instantaneous acceleration:
𝑎𝑖𝑛𝑠𝑡 = lim
∆𝑡 → 0
∆𝑣
∆𝑡
𝑑𝑣
= 𝑑𝑡 ; derivative of velocity
Parameters:
derivative
Displacement - velocity - acceleration
Integral
Acceleration - velocity - displacement
D. Special Cases
1. Constant Velocity
- Uniform motion
- Zero acceleration
𝑥 = 𝑥0 + 𝑣𝑡
*Instantaneous velocity = average velocity
𝑣=
𝑥−𝑥0
𝑡−0
= 𝑣𝑎𝑣𝑒
2. Constant acceleration
𝑣 = 𝑣𝑜 + 𝑎𝑡
2
𝑥 = 𝑥𝑜 + 𝑣𝑜𝑡 +
2
2
𝑣𝑓 = 𝑣𝑖 + 2𝑎𝑥
𝑎𝑡
2
Graphs
Position vs. time
b.) constant velocity
Wherein:
Velocity: slope of the tangent line
Direction of velocity: orientation of slope
Speed: Magnitude of slope
Velocity vs. time
Displacement is linear while acceleration is
0
C. Constant acceleration
Wherein:
Acceleration: slope of the tangent line
Magnitude of acceleration: magnitude of
slope
Direction of acceleration: Orientation of
slope
2. Kind of motions
a.) Zero Velocity
Wherein velocity and acceleration is 0
D. Free fall
- Sole influence of gravity
- Constant acceleration
Formulae
2
𝑦 = 𝑦𝑜 + 𝑣𝑜𝑡 −
𝑔𝑡
2
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𝑣 = 𝑣𝑜 − 𝑔𝑡
2
2
𝑣 = 𝑣0 − 2𝑔𝑥
MODULE 2: Kinematics in 2D and 3D
MODULE 1: COMPILATION OF
FORMULAE
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒: 𝐴 + 𝐵 = 𝐶
𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡: 𝐴 − 𝐵 = 𝐶, 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
Displacement: express position of a
particle in space
𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘
Average velocity
𝑟 −𝑟
∆𝑟
𝑣𝑎𝑣 = 𝑡2−𝑡 1 = ∆𝑡
2
1
𝑆𝑎𝑣𝑒 =
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑡𝑖𝑚𝑒
Instantaneous velocity
𝑉𝑎𝑣𝑒 =
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑡𝑖𝑚𝑒
𝑣 = 𝑑𝑡 = 𝑑𝑡 𝑖 + 𝑑𝑡 𝑗 + 𝑑𝑡 𝑘
∆𝑥
∆𝑡
𝑣𝑖𝑛𝑠𝑡 = lim
∆𝑡 → 0
𝑑𝑟
𝑑𝑥
= 𝑑𝑡
𝑣𝑓−𝑣𝑖
𝑎𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =
∆𝑡
∆𝑣
∆𝑡
𝑎𝑖𝑛𝑠𝑡 = lim
∆𝑡 → 0
𝑑𝑣
= 𝑑𝑡
𝑑𝑥
At every point along the path of the
motion of the particle, the direction
of the instantaneous velocity is
tangent to the path at that point
Average acceleration
𝑣 −𝑣
𝑡−0
𝑎=
-
= 𝑣𝑎𝑣𝑒
Constant acceleration
𝑣 = 𝑣𝑜 + 𝑎𝑡
2
𝑥 = 𝑥𝑜 + 𝑣𝑜𝑡 +
2
𝑎𝑡
2
2
𝑣𝑓 = 𝑣𝑖 + 2𝑎𝑥
𝑥 = 𝑥0 +
∆𝑣
𝑎𝑎𝑣𝑒 = 𝑡2−𝑡 1 = ∆𝑡
1
Instantaneous acceleration
Constant velocity
𝑥 = 𝑥0 + 𝑣𝑡
𝑣=
𝑑𝑧
-
2
𝑥−𝑥0
𝑑𝑦
𝑣−𝑣𝑜𝑡
-
𝑑𝑣𝑥
𝑖+
𝑑𝑡
𝑑𝑣𝑦
𝑗+
𝑑𝑡
𝑑𝑣𝑧
𝑑𝑡
𝑘
Acceleration parallel to velocity
changes magnitude (x-axis)
Acceleration perpendicular to
velocity changes direction
A. Projectile Motion
Equations
X-component
Constant velocity
Y-component
Freefall
(constant acceleration)
𝑎𝑥 = 0
𝑎𝑦 =− 𝑔
𝑣𝑜𝑥 = 𝑣𝑜𝑐𝑜𝑠θ𝑜
𝑣𝑜𝑦 = 𝑣𝑜𝑠𝑖𝑛θ𝑜
𝑣𝑥 = 𝑣𝑜𝑥
𝑣𝑦 = 𝑣𝑜𝑦 − 𝑔𝑡
𝑥 = 𝑥𝑜 + 𝑣𝑜𝑥𝑡
𝑦 = 𝑦𝑜 + 𝑣𝑜𝑦𝑡 − 2 𝑔𝑡
2
2
𝑥 = 𝑥𝑜 + 𝑣𝑓𝑡 +
𝑎𝑡
2
Free fall
2
𝑦 = 𝑦𝑜 + 𝑣𝑜𝑡 −
2
𝑑 = 𝑣0𝑡 +
𝑎𝑡
2
𝑣 = 𝑣𝑜 − 𝑔𝑡
2
2
𝑣 = 𝑣0 − 2𝑔𝑥
𝑔𝑡
2
1
2
-
-
Parallel to the velocity
Change speed
Trajectory of a projectile motion
𝑔
𝑥−𝑥
2
𝑜
𝑦 = 𝑦𝑜 + (𝑥 − 𝑥𝑜)𝑡𝑎𝑛θ − 2 ( 𝑣 𝑐𝑜𝑠θ
)
Reference frame
𝑜
Range (horizontal displacement)
2 𝑠𝑖𝑛2θ
𝑅 = 𝑉0 𝑔 , applicable when returning to
starting height
1.) Inertial reference frame
Coordinate system that is stationary
or moving with constant speed
Time of Flight
𝑇=
2𝑣𝑜𝑠𝑖𝑛θ
𝑔
B. Uniform Circular motion UCM
- Constant speed
- Accelerating due to the change of
direction
- Velocity not constant
- Acceleration (towards the center of
the circular path) is perpendicular to
tangential velocity
Radial acceleration
𝑣
∆𝑣 = 𝑅1 ∆𝑠
2
𝑣
𝑎𝑟𝑎𝑑 = 𝑅
Period
𝑇=
2π𝑟
𝑣
Frequency
1
𝑓= 𝑇
B. Non-uniform circular motion
- An object moves in a circle with
variable speed
- Two components:
- Radial acceleration
2
𝑣
𝑎𝑟𝑎𝑑 = 𝑅
-
-
Perpendicular to velocity
Change direction of velocity
-
Tangential acceleration
𝑎𝑡𝑎𝑛 =
𝑑[𝑣]
𝑑𝑡
2.) Non-inertial reference frame
Coordinate system that is
accelerating with respect to an
inertial frame
Relative Motion
- Velocity of moving objects relative to
other moving or stationary objects.
- Use vectors
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