Foundation course - PHYSICS
Lecture 4-2: Kinematics of a particle
● position and displacement
● average velocity, average speed, instantaneous velocity
● average acceleration, instantaneous acceleration
● one-dimensional motion
● free fall, upward and downward throws
Naďa Špačková spackova@physics.muni.cz
Kinematics
Essential questions:
● How fast and how far an object moves
● In which direction the object is moving
● Whether the object is speeding up or slowing down
● Whether the object is standing still or moving at a constant speed
Motion:
● Objects move in many different ways
● The least complicated motion is a movement along a straight line
● Description of motion is a description of place and time
● Other characteristics of motion are velocity (speed) and acceleration
Coordinate systems
● gives the location of the zero point of the variable you are
studying and the direction in which the values of the variable
increase
● the origin = the point with zero value of all variables
● the position is described by the distance and direction
● vectors = quantities characterized by both magnitude (size)
and direction
● scalars = quantities characterized only by its size
The red vector has the same magnitude as
the green vector, but opposite direction.
Blue arrows represent the same vector.
Particle Models
● the object of interest is replaced with a single point
● the object’s size must be much less than the distance it moves
● the object’s internal motions are ignored
Position and displacement
0-1-2 -1-3 321
origin
x(m)
positive direction
negative direction
Vectors and scalars
Δt = tfinal−tinitial
Δ x = xfinal−xinitial
● time intervals = scalars
● time interval
● positions and displacements = vectors
● displacement
● vector addition and subtraction:
ri
rf
Δ r
xi
xf
Δ x
One-dimensional situation:
Two-dimensional situation:
Δ x = xf −xi xf = Δ x + xi
R = A – B = A + (-B) Δr = rf −ri
Position-time graph
1.0 2.0 3.0 4.0 5.0
1.0
2.0
3.0
Time (s)
Position(m)
Position-time graph
Example 1: motion in negative direction
Example 2: two motions in positive direction,
the same speed, different initial position
1.0 2.0 3.0 4.0 5.0
1.0
2.0
3.0
Time (s)
Position(m)
-1.0
-2.0
1.0 2.0 3.0 4.0 5.0
1.0
2.0
3.0
Time (s)
Position(m)
-1.0
-2.0
Average velocity and average speed
1.0 2.0 3.0 4.0 5.0
1.0
2.0
3.0
Time (s)
Position(m)
-1.0
-2.0
For a fixed time interval the
magnitude of the displacement is
greater for the cyan object: this
object is moving faster
¯v≡
Δ x
Δt
=
xfinal−xinitial
tfinal−tinitial
Average velocity
Average speed
vavg =
totaldistance
Δt
Velocity (speed) units: 1 m.s-1
Caution:
● average velocity = vector
● average speed = scalar
Examples: average velocity and average speed
An automobile travels on a straight road for 40 km at 30 km/h. It then continues in the same direction
for another 40 km at 60 km/h. (a) What is the average velocity of the car during the full 80 km trip?
(Assume that it moves in the positive x direction.) (b) What is the average speed?
An automobile travels on a straight road for 40 km at 30 km/h. It then continues in the opposite
direction for another 40 km at 60 km/h. (a) What is the average velocity of the car during the full 80 km
trip? (b) What is the average speed?
Instantaneous velocity and speed
Instantaneous velocity
v=lim
Δt→0
Δ x
Δt
=
d x
d t
r
tangent
trajectory
v
Two-dimensional situation:
Velocity vector is always tangent to the
particle’s path at the particle’s position.
Motion along the straight line:
instantaneous velocity is of the
same direction as displacement
x
v
Equation of motion
1.0 2.0 3.0 4.0 5.0
1.0
2.0
3.0
Time (s)
Position(m)
-1.0
-2.0
General representation of the linear
function:
y...quantity on the vertical axis
m...line’s slope
x… quantity on the horizontal axis
b...line’s y-intercept
x=¯v t+xi
} xi
α
tan(α)=
Δ x
Δt
y = mx+b
Equation of motion for a position v. time graph:
Uniform and nonuniform motion
Uniform motion
● moving along the a straight line with an unchanging velocity
Nonuniform motion
● velocity is changing
Acceleration
is the rate at which the object’s velocity changes
Acceleration is a vector (magnitude, direction)
a
Δv
a
Δv
v
v
Velocity-time graphs
1.0 2.0 3.0 4.0 5.0
1.0
2.0
3.0
Time (s)
Velocity(m/s)
-1.0
-2.0
α
tan(α)=
Δ v
Δt
The slope of v(t) represents acceleration
Velocity(m/s)
Time (s)
A
C
B
D
E
Examples of various accelerations:
a = zero (A, E)
a = positive (B, D)
a = negative (C)
Acceleration
¯a≡
Δ v
Δt
=
vfinal−vinitial
tfinal−tinitial
Instantaneous acceleration
a=lim
Δ t→0
Δ v
Δt
=
d v
d t
Average acceleration
1.0 2.0 3.0 4.0 5.0
1.0
2.0
3.0
Time (s)Velocity(m/s)
-1.0
-2.0
Example: Non-constant acceleration
Acceleration with constant speed
acceleration is associated with a change in the direction of motion
v
v
Acceleration units: 1 m.s-2
Motion with constant acceleration
Time
Velocity
Area under the graph represents
displacement of the particle
Time
Position
Slope varies
Slope = a
Time
Acceleration
Slope = 0
Checkpoint question:
Match each vx
-t graph with the ax
-t
graph that best describes the motion.
Velocity with average acceleration
tinitial
tfinal
vinitial
vfinal
Time (s)
Velocity(m/s)
¯a≡
Δ v
Δt
Δ v = ¯a Δt
vf −vi = ¯a Δt
vf = vi + atf
When acceleration is constant, the
average acceleration is the same as
the instantaneous acceleration.
Motion with an initial nonzero velocity
Δ x=Δ xrectangle+Δ xtriangle=vi Δt+
1
2
Δ v Δt=vi Δt+
1
2
a(Δt)
2
xf −xi=vi Δt+
1
2
a(Δt)
2
xf =xi+vi tf +
1
2
atf
2
if ti
= 0
vf = vi + ¯a Δt
v = v0 + at
x − x0 = v0 t +
1
2
at
2
x − x0, v0, v ,a,t
Examples: equations of motion
A car accelerates from rest with a uniform acceleration. After traveling a distance of
160 m its velocity is 26 m/s. What is its acceleration?
Free fall
Free fall is the motion of an object when gravity is the only significant force
acting on it.
Galileo’s experiments with
free fall motion
g
ay =−g
vy=−gt
y = h −
1
2
gt2
y
x
h
v0y
= 0
Free fall
Free-fall acceleration:
● near Earth’s surface is about 9.8 m/s2
downward (each second
velocity increases by 9.8 m/s)
● it is not dependent on mass, density and shape of the falling object
● its magnitude depends on distance from the Earth
Without air friction (in vacuum) all objects
are falling equally.
Downward throw
g
y
x
h
v0y
≠ 0
v0y
vy =−v0 y − gt
y = h − v0 y t −
1
2
gt2
Downward throw
● initial velocity is not zero and is in the same direction as free fall acceleration
Upward throw
Upward throw
● initial velocity is not zero and is in the opposite direction as free fall acceleration
g
x
ymax
v0y
≠ 0
v0y
vy = v0 y − gt
y = y0 + v0 y t −
1
2
gt2
Summary
vf = vi + atf xf =xi+vi tf +
1
2
atf
2
Motion with constant acceleration
Time
Position
Velocity
vi
xi