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SCIENCE – PHYSICSSTEM GARAGE
Linear Motion
Chin-Sung Lin
SCIENCE – PHYSICSSTEM GARAGE
Relative Motion
SCIENCE – PHYSICSSTEM GARAGE
Relative Motion
Motion is Relative
q All motion is relative
q Motion is meaningless unless we specify motion relative to a specific object
q Each such choice is known as a frame of reference
SCIENCE – PHYSICSSTEM GARAGE
Relative Motion
Frame of Reference
q An arbitrary coordinate system with reference to which the position or motion of something is described or physical laws are formulated
SCIENCE – PHYSICSSTEM GARAGE
Relative Motion
Frame of Reference
q Two frames of reference moving relative to each other with a constant velocity are completely equivalent, and the same laws of mechanics hold in both
SCIENCE – PHYSICSSTEM GARAGE
Distance & Displacement
SCIENCE – PHYSICSSTEM GARAGE
Distance
A distance (d) is a scalar quantity A distance is a numerical description of how far
apart objects are or the initial to the final position of an object
Unit: meter (m)
Distance
SCIENCE – PHYSICSSTEM GARAGE
Displacement
A displacement (d) is a vector quantity A displacement represents the length and direction
of the shortest distance from the initial to the final position of an object
Its direction is from the initial location to the final location
Unit: meter (m)
Displacement
SCIENCE – PHYSICSSTEM GARAGE
Speed & Velocity
SCIENCE – PHYSICSSTEM GARAGE
Speed
A speed (v) is a scalar quantity
Distance (d) covered per unit of time (t). Speed is a measure of how fast something is moving. It is the rate at which distance is covered
Unit: meters per second (m/s), miles per hour (mi/h), kilometers per hour (km/h)
SCIENCE – PHYSICSSTEM GARAGE
Average Speed
Average speed (v):
v = d / t
d = total distance covered (m)
t = time interval (s)
t
d slope = v
SCIENCE – PHYSICSSTEM GARAGE
Instantaneous Speed
The speed (v) at any instance of an object is called the instantaneous speed
It is equal to the slope of the tangent line at that moment
t
d slope = v
SCIENCE – PHYSICSSTEM GARAGE
Approximation of Instantaneous Speed
SCIENCE – PHYSICSSTEM GARAGE
Constant Speed
The speed at any instance of an object is constant
t
d
slope = vx
t
v
slope = 0
vx
SCIENCE – PHYSICSSTEM GARAGE
Constant Speed
The speed at any instance of an object is constant
t
d slope = vx
t
v
slope = 0
vx
SCIENCE – PHYSICSSTEM GARAGE
Constant Speed
The speed at any instance of an object is constant
t
d slope = 0
t
v
slope = 0
0
SCIENCE – PHYSICSSTEM GARAGE
Velocity
A velocity (v) is a vector quantity Velocity is the measurement of the rate and
direction of change in the displacement (d) of an object
v = d / t The speed is the magnitude of velocity
Unit: meters per second (m/s), miles per hour (mi/h), kilometers per hour (km/h)
SCIENCE – PHYSICSSTEM GARAGE
Speed & Velocity
Speed has only magnitude Velocity has magnitude and direction
SCIENCE – PHYSICSSTEM GARAGE
Average Velocity
Average velocity (v):
v = d / t
d = total displacement (m)
t = time interval (s)
t
d slope = v
SCIENCE – PHYSICSSTEM GARAGE
Instantaneous Velocity
The velocity (v) at any instance of an object is called the instantaneous velocity
It is equal to the slope of the tangent line at that moment
t
d slope = v
SCIENCE – PHYSICSSTEM GARAGE
Constant Velocity
To have a constant velocity requires both constant speed and constant direction
Motion at constant velocity is motion in a straight line at constant speed
t
d
slope = vx
t
v
slope = 0
vx
SCIENCE – PHYSICSSTEM GARAGE
Constant Velocity
The velocity at any instance of an object is constant
t
d slope = vx
t
v
slope = 0
vx
SCIENCE – PHYSICSSTEM GARAGE
Constant Velocity
The velocity at any instance of an object is constant
t
d slope = 0
t
v
slope = 0
0
SCIENCE – PHYSICSSTEM GARAGE
Constant Velocity
The velocity at any instance of an object is constant
t
d
slope = vx
t
v
slope = 0
vx
SCIENCE – PHYSICSSTEM GARAGE
Constant Velocity
The velocity at any instance of an object is constant
t
d
slope = vx t
v
slope = 0
vx
SCIENCE – PHYSICSSTEM GARAGE
Change Velocity
Either speed or direction (or both) is changing and then the velocity is changing
SCIENCE – PHYSICSSTEM GARAGE
Change Velocity
Motion at constant speed can have changing velocity all the time when it moves along a curved path
Constant speed
Changing velocity
SCIENCE – PHYSICSSTEM GARAGE
Change Velocity
Car example:
In a car there are three controls that are used to change the velocity: the gas pedal, the brake and the steering wheel
SCIENCE – PHYSICSSTEM GARAGE
Acceleration
SCIENCE – PHYSICSSTEM GARAGE
Acceleration
An acceleration (a) is a vector quantity acceleration is the rate of change of velocity with time
a = Δv / t
a = acceleration (m/s2) Δv = change of velocity (m/s) t = time interval (s)
SCIENCE – PHYSICSSTEM GARAGE
Acceleration
Acceleration: the slope of the velocity-time (v-t) graph
t
v
slope = ax (accelera5on)
SCIENCE – PHYSICSSTEM GARAGE
Constant Acceleration
The acceleration at any instance of an object is constant. In high school physics. We only deal with constant
acceleration.
t
v
slope = ax
t
a
slope = 0
ax
SCIENCE – PHYSICSSTEM GARAGE
Constant Acceleration
The acceleration in the negative direction is also called deceleration. In high school physics, we only deal with constant acceleration.
t
v
slope = ax
t
a
slope = 0
ax
SCIENCE – PHYSICSSTEM GARAGE
Constant Acceleration
The acceleration at any instance of an object is constant. Initial velocity is negative.
t
v slope = ax
t
a
slope = 0
ax
vi
Ini5al velocity is nega5ve
velocity is zero
SCIENCE – PHYSICSSTEM GARAGE
Constant Acceleration
The acceleration in the negative direction is also called deceleration.
Initial velocity is positive.
t
v
slope = ax
t
a
slope = 0
ax
Ini5al velocity is posi5ve
velocity is zero
SCIENCE – PHYSICSSTEM GARAGE
Displacement, Velocity & Acceleration
d-t
t
v slope = ax
t
aslope = 0
ax
t
d slope = increasing at constant rate
v-t a-t
SCIENCE – PHYSICSSTEM GARAGE
Displacement, Velocity & Acceleration
d-t
t
v
slope = ax t
a
slope = 0
ax t
d slope = decreasing at constant rate
v-t a-t
SCIENCE – PHYSICSSTEM GARAGE
Change Acceleration
Either speed or direction (or both) is changing, i.e., changes in the state of motion, and then the acceleration is changing
The acceleration applies to increases as well as decreases in speed
The decrease in speed is also called deceleration, or negative acceleration
SCIENCE – PHYSICSSTEM GARAGE
Acceleration An acceleration (a) is also a scalar quantity
When linear (straight-line) motion is considered, it is common to use speed and velocity interchangeably and the acceleration may be expressed as the rate at which speed changes
a = Δv / t
a = acceleration (m/s2) Δv = change of speed (m/s) t = time interval (s)
SCIENCE – PHYSICSSTEM GARAGE
Acceleration
Car example:
Cars having good acceleration means being able to change velocity quickly and does not necessarily refer to how fast something is moving
SCIENCE – PHYSICSSTEM GARAGE
Free Fall
SCIENCE – PHYSICSSTEM GARAGE
Free Fall
When there is no air resistance and the gravity is the only thing affecting a falling object
SCIENCE – PHYSICSSTEM GARAGE
Elapsed Time
The elapsed time is the time that has elapsed, or passed, since the beginning of the fall
SCIENCE – PHYSICSSTEM GARAGE
Acceleration due to Gravity
Acceleration due to gravity (g): The free falling object is experiencing acceleration, i.e., a change in speed
The value of acceleration (g) is about 10 m/s2. More accurately, g is 9.81 m/s2
The speed increase 9.81 m/s per second
g is 9.81 m/s2
SCIENCE – PHYSICSSTEM GARAGE
Acceleration due to Gravity
0 s
1 s 5 s
2 s 4 s
3 s
6 s
7 s
v = 30 m/s
v = 20 m/s
v = 10 m/s
v = 0 m/s
v = -‐40 m/s
v = -‐30 m/s
v = -‐20 m/s
v = -‐10 m/s
v = 0 m/s
SCIENCE – PHYSICSSTEM GARAGE
Acceleration due to Gravity
t
a
Constant Accelera5on
g
a-t
g = -‐9.81 m/s2
SCIENCE – PHYSICSSTEM GARAGE
g = -‐9.81 m/s2
Velocity Change due to Gravity
t
v
slope = g
v-t v = 0
SCIENCE – PHYSICSSTEM GARAGE
Displacement Change due to Gravity
d-t
t
d slope decreasing at constant rate dy
v = 0
g = -‐9.81 m/s2
d = dy
d = 0
SCIENCE – PHYSICSSTEM GARAGE
Displacement, Velocity & Acceleration due to Gravity
d-t
t
v
slope = g t
a
slope = 0
g t
d slope decreasing at constant rate
v-t a-t
dy
SCIENCE – PHYSICSSTEM GARAGE
Displacement, Velocity & Acceleration due to Gravity
d-t
t
v
slope = g t
a
slope = 0
g t
d slope decreasing at constant rate
v-t a-t
dy
Constant Direct Propor5on
Quadra5c Parabola
SCIENCE – PHYSICSSTEM GARAGE
Displacement, Velocity & Acceleration due to Gravity
d-t
t
v slope = g
t
a
slope = 0
g
t
d slope increasing at constant rate
v-t a-t
dy
Slope of v-‐t Area under v-‐t
SCIENCE – PHYSICSSTEM GARAGE
Instantaneous Velocity in Free Falling
t
v
slope = g
v-t
t = tf
v = vf
vf = ?
SCIENCE – PHYSICSSTEM GARAGE
Instantaneous Velocity in Free Falling
t
v
slope = g
v-t
t = tf
v = vf
vf = ?
v = gt
vf = gtf
SCIENCE – PHYSICSSTEM GARAGE
Instantaneous Velocity in Free Falling
t
v
slope = g
v-t
t = tf
v = vf
The instantaneous velocity of an object falling from rest:
instantaneous velocity = acceleration x elapsed time
v = gt
SCIENCE – PHYSICSSTEM GARAGE
Instantaneous Velocity in Free Falling
t
v
slope = g
v-t t = tf
v = vf
When the falling object has initial velocity (vi), whether the object is moving upward or downward, the acceleration due to gravity is always the same (–9.81 m/s2) the entire time
The final velocity (vf), the instantaneous velocity, is:
vf = vi + gt v = vi
SCIENCE – PHYSICSSTEM GARAGE
Instantaneous Velocity in Free Falling
t
v
slope = g
v-t t = tf
v = vf
When the falling object has initial velocity (vi), whether the object is moving upward or downward, the acceleration due to gravity is always the same (–9.81 m/s2) the entire time
The final velocity (vf), the instantaneous velocity, is:
vf = vi + gt g = (vf - vi) / t
v = vi
SCIENCE – PHYSICSSTEM GARAGE
Instantaneous Velocity in Free Falling
t
v
slope = g
v-t t = tf
v = vf
When the falling object has initial velocity (vi), whether the object is moving upward or downward, the acceleration due to gravity is always the same (–9.81 m/s2) the entire time
The final velocity (vf), the instantaneous velocity, is:
vf = vi + gt g = (vf - vi) / t g = Δv / t
v = vi
SCIENCE – PHYSICSSTEM GARAGE
Average velocity in Free Falling
t
v
slope = g
v-t
tf
vf
For any object moving in linear motion with constant acceleration, the average velocity (v) is:
average velocity = (initial velocity + final velocity) / 2
0
vi
v
v = vi + vf 2
SCIENCE – PHYSICSSTEM GARAGE
Displacement-Time Formula Given initial velocity (vi) and time (t), and based on the
formulas we have gotten, find the displacement-time formula
v = vi + vf
2
vf = vi + gt
v = d / t
SCIENCE – PHYSICSSTEM GARAGE
Displacement-Time Formula Given initial velocity (vi) and time (t), and based on the
formulas we have gotten, find the displacement-time formula
v = vi + vf
2
vf = vi + gt
v = d / t
d = vit + ½ gt2
SCIENCE – PHYSICSSTEM GARAGE
Displacement Change due to Gravity
d-t
t
d Parabola dy
v = 0
g = -‐9.81 m/s2
d = dy
d = 0
d = dy + vit + ½ gt2
SCIENCE – PHYSICSSTEM GARAGE
Distance Change due to Gravity
d-t
t
d Parabola
v = 0
g = 9.81 m/s2
d = 0
d = dy
d = vit + ½ gt2
SCIENCE – PHYSICSSTEM GARAGE
Displacement-Velocity Formula Given initial velocity (vi) and displacement (d), and based
on the formulas we have gotten, find the displacement-velocity formula
d = vit + ½ gt2
g = (vf - vi) / t
SCIENCE – PHYSICSSTEM GARAGE
Displacement-Velocity Formula Given initial velocity (vi) and displacement (d), and based
on the formulas we have gotten, find the displacement-velocity formula
d = vit + ½ gt2
g = (vf - vi) / t vf
2 = vi2 + 2gd
SCIENCE – PHYSICSSTEM GARAGE
Summary of Free Falling Formulas
v = vi + vf
2
vf = vi + gt v = d / t
d = vit + ½ gt2 g = Δv / t
vf2 = vi
2 + 2gd
SCIENCE – PHYSICSSTEM GARAGE
Summary of Linear Motion Formulas
v = vi + vf
2
vf = vi + at v = d / t
d = vit + ½ at2 a = Δv / t
vf2 = vi
2 + 2ad
SCIENCE – PHYSICSSTEM GARAGE
The End