Chapter 2
Motion in One Dimension
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2.1
2.2
2.3
2.4
2.5
Position, Velocity, and Speed
Instantaneous Velocity and Speed
Acceleration
Freely Falling Objects
Kinematic Equations Derived from Calculus.
Kinematics
 Kinematics describes motion while ignoring
the agents that caused the motion
 For now, we will consider motion in one
dimension
 Along a straight line
 We will use the particle model
 A particle is a point-like object, that has mass but
infinitesimal size
Position
 Position is defined in terms
of a frame of reference
 For one dimension the
motion is generally along
the x- or y-axis
 The object’s position is its
location with respect to
the frame of reference
Position-Time Graph
 The position-time
graph shows the
motion of the
particle (car)
 The smooth curve is
a guess as to what
happened between
the data points
Displacement
 Displacement is defined as the change in
position during some time interval
 Represented as x
 x = xf - xi
 SI units are meters (m), x can be positive or
negative
 Displacement is different than distance.
Distance is the length of a path followed by a
particle.
Vectors and Scalars
 Vector quantities that need both magnitude (size or numerical value) and
direction to completely describe them
 We will use + and – signs to indicate vector
directions
 Scalar quantities are completely described
by magnitude only
Average Velocity

The average velocity is the rate at which the
displacement occurs
v average
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

 x x f  xi


t
t
The dimensions are length / time [L/T]
The SI units are m/s
Is also the slope of the line in the position –
time graph
Average Speed
 Speed is a scalar quantity
same units as velocity
total distance / total time
 The average speed is not (necessarily)
the magnitude of the average velocity
Instantaneous Velocity
 Instantaneous velocity is the limit of the
average velocity as the time interval becomes
infinitesimally short, or as the time interval
approaches zero
 The instantaneous velocity indicates what is
happening at every point of time
Instantaneous Velocity
 The general equation for instantaneous
velocity is
 x dx
v x  lim

dt
t  0  t
 The instantaneous velocity can be
positive, negative, or zero
Instantaneous Velocity
 The instantaneous
velocity is the slope
of the line tangent to
the x vs t curve
 This would be the
green line
 The blue lines show
that as t gets
smaller, they
approach the green
line
Instantaneous Speed
 The instantaneous speed is the
magnitude of the instantaneous velocity
 Remember that the average speed is
not the magnitude of the average
velocity
Average Acceleration
 Acceleration is the rate of change of the
velocity
 v x v xf  v xi
ax 

t
t
 Dimensions are L/T2
 SI units are m/s²
Instantaneous Acceleration
 The instantaneous acceleration is the limit of
the average acceleration as t approaches 0
 v x dv x d x
a x  lim

 2
t  0  t
dt
dt
2
Instantaneous Acceleration
 The slope of the
velocity vs. time
graph is the
acceleration
 The green line
represents the
instantaneous
acceleration
 The blue line is the
average
acceleration
Acceleration and Velocity
 When an object’s velocity and acceleration
are in the same direction, the object is
speeding up
 When an object’s velocity and acceleration
are in the opposite direction, the object is
slowing down
Acceleration and Velocity
 The car is moving with constant positive
velocity (shown by red arrows maintaining the
same size)
 Acceleration equals zero
Acceleration and Velocity
 Velocity and acceleration are in the same direction
 Acceleration is uniform (blue arrows maintain the
same length)
 Velocity is increasing (red arrows are getting longer)
 This shows positive acceleration and positive velocity
Acceleration and Velocity
 Acceleration and velocity are in opposite directions
 Acceleration is uniform (blue arrows maintain the
same length)
 Velocity is decreasing (red arrows are getting
shorter)
 Positive velocity and negative acceleration
1D motion with constant
acceleration
aa
v f  vi
t f  ti
tf – ti = t
v f  v i  at
1D motion with constant acceleration
 In a similar manner we can rewrite equation
for average velocity:
v avg  v 
x f  xi
t f  ti
 and than solve it for xf
x f  x i  v a vg t
1
 Rearranging, and assuming v
(v0  v f )
avg 
2
1D motion with constant acceleration
(1)
Using
v f  v i  at
(1)
and than substituting into equation for final
position yields
(2)
1 2
x f  x i  v 0 t  at
2
(2)
Equations (1) and (2) are the basic kinematics
equations
1D motion with constant acceleration
These two equations can be combined to
yield additional equations.
We can eliminate t to obtain
v  v  2 a ( x f  xi )
2
f
2
i
Second, we can eliminate the acceleration a
to produce an equation in which acceleration
does not appear:
1
x f  x 0  ( v 0  v f )t
2
Kinematics with constant acceleration
- Summary
v f  vi  at
vi  vf
v
2
1
2
Δx  v i t  2 at
v f  v  2ax
2
2
i
Kinematic Equations - summary
Kinematic Equations
 The kinematic equations may be used to
solve any problem involving one-dimensional
motion with a constant acceleration
 You may need to use two of the equations to
solve one problem
 Many times there is more than one way to
solve a problem
Kinematics - Example 1
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How long does it take for a train to come to rest if
it decelerates at 2.0m/s2 from an initial velocity of
60 km/h?
A car is approaching a hill at 30.0 m/s when its engine
suddenly fails just at the bottom of the hill. The car moves
with a constant acceleration of –2.00 m/s2 while coasting up
the hill. (a) Write equations for the position along the slope
and for the velocity as functions of time, taking x = 0 at the
bottom of the hill, where vi = 30.0 m/s. (b) Determine the
maximum distance the car rolls up the hill.
Graphical Look at Motion:
displacement-time curve
 The slope of the
curve is the velocity
 The curved line
indicates the
velocity is changing
 Therefore, there is
an acceleration
Graphical Look at Motion:
velocity-time curve
 The slope gives the
acceleration
 The straight line
indicates a constant
acceleration
Graphical Look at Motion:
acceleration-time curve
 The zero slope
indicates a constant
acceleration
Freely Falling Objects
 A freely falling object is any object
moving freely under the influence of
gravity alone.
 It does not depend upon the initial
motion of the object
Dropped – released from rest
Thrown downward
Thrown upward
Acceleration of Freely Falling Object
 The acceleration of an object in free fall is
directed downward, regardless of the initial
motion
 The magnitude of free fall acceleration is
g = 9.80 m/s2



g decreases with increasing altitude
g varies with latitude
9.80 m/s2 is the average at the Earth’s surface
Acceleration of Free Fall
 We will neglect air resistance
 Free fall motion is constantly accelerated
motion in one dimension
 Let upward be positive
 Use the kinematic equations with
ay = g = -9.80 m/s2
Free Fall Example
 Initial velocity at A is upward (+)
and acceleration is g (-9.8 m/s2)
 At B, the velocity is 0 and the
acceleration is g (-9.8 m/s2)
 At C, the velocity has the same
magnitude as at A, but is in the
opposite direction
A student throws a set of keys vertically upward to her
sorority sister, who is in a window 4.00 m above. The keys
are caught 1.50 s later by the sister's outstretched hand.
(a) With what initial velocity were the keys thrown? (b)
What was the velocity of the keys just before they were
caught?
A ball is dropped from rest from a height h above the
ground. Another ball is thrown vertically upwards from the
ground at the instant the first ball is released. Determine the
speed of the second ball if the two balls are to meet at a
height h/2 above the ground.
A freely falling object requires 1.50 s to travel the last 30.0 m
before it hits the ground. From what height above the ground
did it fall?
Motion Equations from Calculus
 Displacement
equals the area
under the velocity –
time curve
lim
tn 0
v
n
tf
xn
tn   v x (t )dt
ti
 The limit of the sum
is a definite integral
Kinematic Equations – General
Calculus Form
dv x
ax 
dt
t
v xf  v xi   a x dt
0
dx
vx 
dt
t
x f  xi   v x dt
0
Kinematic Equations – Calculus Form
with Constant Acceleration
 The integration form of vf – vi gives
v xf  v xi  a x t
 The integration form of xf – xi gives
1
2
x f  x i  v xi t  a x t
2
The height of a helicopter above the ground is given by
h = 3.00t3, where h is in meters and t is in seconds. After
2.00 s, the helicopter releases a small mailbag. How long
after its release does the mailbag reach the ground?
Automotive engineers refer to the time rate of change of
acceleration as the "jerk." If an object moves in one
dimension such that its jerk J is constant, (a) determine
expressions for its acceleration ax(t), velocity vx(t), and
position x(t), given that its initial acceleration, speed, and
position are axi , vxi, and xi , respectively. (b) Show that
2
ax2  axi
 2 j (vx  vxi )
The acceleration of a marble in a certain fluid is
proportional to the speed of the marble squared, and is
given (in SI units) by a = –3.00 v2 for v > 0. If the marble
enters this fluid with a speed of 1.50 m/s, how long will it
take before the marble's speed is reduced to half of its
initial value?
A test rocket is fired vertically upward from a well. A
catapult gives it initial velocity 80.0 m/s at ground level. Its
engines then fire and it accelerates upward at 4.00 m/s2
until it reaches an altitude of 1 000 m. At that point its
engines fail and the rocket goes into free fall, with an
acceleration of –9.80 m/s2. (a) How long is the rocket in
motion above the ground? (b) What is its maximum
altitude? (c) What is its velocity just before it collides with
the Earth?
An inquisitive physics student and mountain climber
climbs a 50.0-m cliff that overhangs a calm pool of water.
He throws two stones vertically downward, 1.00 s apart,
and observes that they cause a single splash. The first
stone has an initial speed of 2.00 m/s. (a) How long after
release of the first stone do the two stones hit the water?
(b) What initial velocity must the second stone have if
they are to hit simultaneously? (c) What is the speed of
each stone at the instant the two hit the water?