Outline

1
Chapter 4
Motion With a Changing Velocity
2

• P2.27 Find the magnitude and direction of the
  vector with the following components
• x -5.0 cm, y 8.0 cm
• Fx 120 N, Fy -60.0 N
• vx -13.7 m/s, vy -8.8 m/s
• ax 2.3 m/s2, ay 6.5 cm/s2

3

• P2.56 A 3.0 kg block is at rest on a horizontal
  floor. If you push horizontally on the block with
  a force of 12.0 N, it just starts to move.
• What is the coefficient of static friction?
• (b) A 7.0-kg block is stacked on top of the
  3.0-kg block. What is the magnitude F of the
  force acting horizontally on the 3.0-kg block as
  before, that is required to make the two blocks
  start to move?

4
P3.47 A 2010-kg elevator moves with an upward
acceleration of 1.50 m/s2. What is the tension
that supports the elevator?
P3.48 A 2010-kg elevator moves with a downward
acceleration of 1.50 m/s2. What is the tension
that supports the elevator?
5
Kinematic Equations for Const. Acceleration
Fnet ma. If Fnet is const, a will also be
const. Uniformly accelerated motion a const.
6
A car moves at a constant acceleration of
magnitude 5 m/s2. At time t 0, the magnitude of
its velocity is 8 m/s. What is the magnitude of
its velocity at (i) t 2s? (ii) t 4s?
(iii) t 10s?
A car moves at a constant acceleration of
magnitude 5.7 m/s2. At time t 0, the magnitude
of its velocity is 18.3 m/s. What is the
magnitude of its velocity at t 2.2s?
7
Kinematic Equations for Const. Acceleration

• Consider an object on which a net force Fnet acts
  on it. Thus it moves with an acceleration.
• As the object moves, its velocity changes.

Fnet
Fnet
Fnet
a
a
Time 0 Initial position x0 Initial velocity
v0
Time t Final position x Final velocity v
8
Kinematic Equations for Const. Acceleration
Fnet ma. If Fnet is const, a will also be
const. Uniformly accelerated motion a const.

• aave ainst
• Let us use initial time, t1 0.
• Final time, t2 t, hence ?t t 0 t
• Position initial, x1 x0, final, x2 x
• Velocity, initial v1 v0, final, v2 v

9
Kinematic Equations for Constant Acceleration
Uniformly accelerated motion a constant.
Time Initial 0, final t Positions Initial
x0, final x Velocity Initial v0, final v
v v0 at x x0 vot ½ at2 v2 v02
2a(x-x0) Average velocity vav (v0 v)/2
10
Example Problem 4.14

• A train traveling at a constant speed of
• 22 m/s, comes to an incline with a constant
  slope. While going up the incline the train slows
  down with a constant acceleration of magnitude
  1.4 m/s2.
• Draw a graph of vx versus t.
• What is the speed of the train after 8.0s on the
  incline?
• How far has the train traveled up the incline
  after 8.0 s?

11
A car moving south slows down with at a constant
acceleration of 3.0 m/s2. At t 0, its velocity
is 26 m/s. What is its velocity at t 3 s?

1. 35 m/s south
2. 17 m/s south
3. 23 m/s south
4. 29 m/s south
5. 17 m/s north

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12
A car initially traveling at 18.6 m/s begins to
slow down with a uniform acceleration of 3.00
m/s2. How long will it take to come to a stop?

1. 55.8 s
2. 15.6 s
3. 6.20 s
4. 221.6 s
5. None of these

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13
Free Fall

• Free fall Only force of gravity acting on an
  object making it fall.
• Effect of air resistance is assumed negligible.
• Force of gravity acting on an object near the
  surface of the earth is F W mg.
• Acceleration of any object in free fall
• a g 9.8 m/s2 down (ay -9.8 m/s2).

14
Free Fall
ay -9.8 m/s2 ax 0
y
x
15
Free Fall contd
1. a g, regardless of mass of object.
16
2. a g, regardless of initial velocity
ay -9.8 m/s2, ax 0
y
v0 0
v0 -15 m/s
v0 15 m/s
x
17
3. Free Fall Motion is symmetric.
ay -9.8 m/s2, ax 0

• At the maximum height
• vy 0
• Speed at equal heights will be equal.
• Equal time going up and down.

y
v0 5 m/s
x
18
Example Problem 4.32

• A stone is launched straight up by a slingshot.
  Its initial speed is 19.6 m/s and the stone is
  1.5 m above the ground when launched.
• How high above the ground does the stone rise?
• How much time elapses before the
• stone hits the ground?

19
APPARENT WEIGHT
A physics student whose mass is 40 kg stands
inside an elevator on a scale that reads his
weight in Newtons.
Scale Reading Normal force the scale
exerts
on the student.
Scale Reading N mg
40 x 9.8 N
392 N
20
1. Elevator at rest. What will be the scale
reading?
N
Fnet N W may At rest means ay 0. Hence N
W, ie apparent weight true weight 40 x 9.8
392 N
W mg
21
2. Elevator accelerating upwards with ay 2.0
m/s2. What will be the scale reading?
22
2. Elevator accelerating upwards with ay 2.0
m/s2. What will be the scale reading?
Fnet N W may ay 2.0 m/s2 (positive
because acceleration is upwards) . Hence, N W
N mg may. N mg may m(gay) 40(9.8
2.0) 472 N ie, apparent weight is greater
than the true weight.
23
3. Elevator accelerating downwards with ay 2.0
m/s2. What will be the scale reading?
24
3. Elevator accelerating downwards with ay 2.0
m/s2. What will be the scale reading?
Fnet N W may ay - 2.0 m/s2 (negative
because acceleration is downwards) . Hence N W
N mg -may. N mg - may m(g - ay) 40(9.8
- 2.0) 312 N ie, apparent weight is less than
the true weight.
25
A 112.0-kg person stands on a scale inside an
elevator moving downward with an acceleration of
1.80 m/s2. What will be the scale reading?

1. 1299 N
2. 1,098 N
3. 896 N
4. 112 N
5. 0 N

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26
A ball is kicked straight up from ground level
with initial velocity of 22.6 m/s. How high above
the ground will the ball rise?

1. 9.8 m
2. 3.00 m
3. 1.15 m
4. 26.1 m
5. 19.6 m

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27
WEIGHTLESSNESS
If the elevator was going down with an
acceleration ay g -9.8m/s2, then N m(g-g)
0 ie, apparent weight 0 This is
weightlessness or zero gravity Apparent
weight of an object in free fall is zero while
its true weight remains unchanged.
28
Equilibrium
Newtons 2nd Law Fnet ma For an object in
equilibrium Fnet 0 Static (v 0) 0r dynamic
(v constant) eqlbm. 2-dimensions, separate the
x and y components and treat the problem as two
1-dim problems. Fx max Fy may For
equilibrium, ?Fx max 0 and ? Fy may 0
29
2-Dimensions

• X and Y are INDEPENDENT!
• Break 2-D problem into two 1-D problems.

30
Equilibrium
Determine the tension in the 6 m rope if it sags
0.12 m in the center when a gymnast with weight
250 N is standing on it.
y
x direction ?Fx max 0 -TL cosq TR cosq
0 TL TR
TR
TL
x
W
TR
y direction ?Fy may 0 TL sinq TR sinq - W
0 2 T sinq W T W/(2 sinq) 3115 N
.12 m
q
3 m
31
Equilibrium on a Horizontal Plane

• Object at rest or moving with const. velocity
• Fx max 0 and ? Fy may 0

Object at rest
Sliding with constant velocity
No motion until F gt fsmax ?Fx F - fs 0 or F
fs. ?Fy N - W 0 or N W
?Fx F - fk 0 or F fk ?Fy N - W 0 or N
W
32
Equilibrium on an inclined Plane

• An object at rest on an inclined plane
• Fx max 0 and ? Fy may 0

33
Equilibrium on an inclined Plane
An object at rest on an inclined plane ?Fx max
0 and ?Fy may 0
?Fy may 0 N - Wcos? 0 or N Wcos? ?Fx
max 0 fs - Wsin? 0 or fs Wsin?
If angle ? is increased, the object will
eventually slide down the plane. Sliding will
start beyond angle ?max At ?max fsmax
W.sin?max. But fsmax ?sN ?s(Wcos?max) Therefor
e, ?sWcos?max Wsin ?max OR ?s (Wsin?max)/
Wcos?max ie, ?s tan?max
34
Equilibrium on an inclined Plane
An object at rest on an inclined plane
N Wcos? fs Wsin?

• If angle ? is increased, the object will
  eventually
• slide down the plane.
• Sliding will start beyond angle ?max
• At ?max fsmax W.sin?max.
• But fsmax ?sN ?s(Wcos?max)
• Therefore, ?sWcos?max Wsin ?max
• OR ?s (Wsin?max)/ Wcos?max ie, ?s tan?max

35
A mass m being pulled uphill by a force F
y
F
x

• If m 510 kg, ?s 0.42, ?k 0.33, ? 15o
• Find minimum force F needed to start the mass
  moving up.
• If the force in (a) is maintained on the mass,
  what will its acceleration be?

N
W.sin?
fk
?
W.cos?
(c) To move the mass with constant speed, what
must the value of F be?
36
A block is at rest on a flat board. The flat
board is gently tilted. At what angle will the
block start to slide? Assume the coefficient of
static friction (?s) between the block and the
board is 0.48.

1. 0.48o
2. 61.3o
3. 28.7o
4. 25.6o
5. 0.00837o

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37
Position, Velocity and Acceleration

• Position, Velocity and Acceleration are Vectors!
• x and y directions are INDEPENDENT!

y direction
x direction
38
Velocity in Two Dimensions
A ball is rolling on a horizontal surface at 5
m/s. It then rolls up a ramp at a 25 degree
angle. After 0.5 seconds, the ball has slowed to
3 m/s. What is the change in velocity?
x-direction vix 5 m/s vfx 3 m/s cos(25) Dvx
3cos(25)5 -2.28m/s
y-direction viy 0 m/s vfy 3 m/s sin(25) Dvy
3sin(25)1.27 m/s
y
x
3 m/s
5 m/s
39
Acceleration in Two Dimensions
A ball is rolling on a horizontal surface at 5
m/s. It then rolls up a ramp at a 25 degree
angle. After 0.5 seconds, the ball has slowed to
3 m/s. What is the average acceleration? Assume
force of gravity is very small.
y
x-direction
y-direction
x
3 m/s
5 m/s
40

• A wagon of mass 50 kg is being pulled by a force
  F of magnitude 100 N applied through the handle
  at 30o from the horizontal. Ignoring friction,
  find the magnitude of
• the horizontal component of F.
• the horizontal component of acceleration.
• the normal force exerted on the wagon.

41
Projectile Motion

• A projectile An object moving in 2-dimensions
  near the surface of the earth with only the force
  of gravity acting on it.
• Eg golf ball, batted base ball, kicked football,
  soccer ball, bullet, etc.
• Assume no air resistance.
• Assume g -9.8 m/s2 constant.
• We are not concerned with the process that
  started the motion!

42
Free Fall 1-dimensional motion.
ay -9.8 m/s2, ax 0
y

• At the maximum height
• vy 0
• Speeds at equal heights will be equal.
• Equal time going up/down.

v0 5 m/s
x
43
PROJECTILE Free Fall motion in 2-dimensions.
ay -9.8 m/s2, ax 0
y
v0 5 m/s
v0y
?
x
v0x
44
PROJECTILE Free Fall motion in 2-dimensions.
ay -9.8 m/s2 ax 0 v0x v0cos? v0y v0sin?
What will happen to the y-component of the
velocity? What will happen to the x-component of
the velocity?
45
Kinematics in Two Dimensions

• x x0 v0xt ½ axt2
• vx v0x axt
• vx2 v0x2 2ax (x - x0)

• y y0 v0yt ½ ayt2
• vy v0y ayt
• vy2 v0y2 2ay (y y0)

x and y motions are independent! They share a
common time t.
46
Kinematics for Projectile Motionax 0
ay -g

• y y0 v0yt - 1/2 gt2
• vy v0y - gt
• vy2 v0y2 - 2g ?y

• x x0 v0t
• vx v0x

X
Y
47
PROJECTILE Free Fall motion in 2-dimensions.
ay -9.8 m/s2, ax 0
Once the projectile is in air, the only force
acting on it is gravity. Its trajectory (path of
motion) is a parabola. Fnet ma -mg ay -9.8
m/s2 ax 0
48
PROJECTILE Free Fall motion in 2-dimensions.
ay -9.8 m/s2 and ax 0 v0x v0cos? and v0y
v0sin?
49
Two balls A and B of equal mass m. Ball A is
released to fall straight down from a height h.
Ball B is thrown horizontally. Which ball lands
first?
A
B
h
ay -9.8 m/s2 ax 0 Vo 0 v0x 0 v0y 0
ay -9.8 m/s2 ax 0 Vo? 0 v0x Vo v0y 0
50
A

• y y0 v0yt ½ ayt2
• vy v0y ayt
• vy2 v0y2 2ay (y y0)
• To find time t, use
• y y0 v0yt ½ ayt2
• -h 0 (0 . t) ½ (-g)t2
• Gives 2h gt2 and t ?(2h/g)

ay -9.8 m/s2 ax 0 vo 0 v0x 0 v0y 0 y0
0, y -h
51
B
ay -9.8 m/s2 ax 0 Vo? 0 v0x vo v0y 0

• y y0 v0yt ½ ayt2
• vy v0y ayt
• vy2 v0y2 2ay (y y0)
• To find time t, use
• y y0 v0yt ½ ayt2
• -h 0 (0 . t) ½ (-g)t2
• Gives 2h gt2 and t ?(2h/g)

52
A flatbed railroad car is moving along a track at
constant velocity. A passenger at the center of
the car throws a ball straight up. Neglecting
air resistance, where will the ball land?1.
Forward of the center of the car 2. At the center
of the car 3. Backward of the center of the car
53
Since no air resistance is present, the ball and
the train would be moving with the same
horizontal velocity, and when the ball is tossed,
it is given an additional velocity component in
the vertical direction, but the original
horizontal velocity component remains unchanged,
and lands in the center of the train.
54
P 4.22 A penny is dropped from the observation
deck of the Empire State building (369 m above
the ground). With what velocity does it strike
the ground? Ignore air resistance.
55
P 4.36 An arrow is shot into the air at an angle
of 60.0o above the horizontal with a speed of
20.0 m/s. (a) What are the x- and y-components
of the velocity of the arrow 3.0 s after it
leaves the bowstring? (b) What are the x- and y-
components of the displacement of the arrow
during the 3.0-s interval?
56
A ball is thrown with a speed of 40.0 m/s at 55o
above the horizontal. At the maximum height, its
speed will be

1. 22.9 m/s
2. -9.8 m/s
3. 0 m/s
4. 32.8 m/s
5. 40.0 m/s

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A ball is thrown with a speed of 40.0 m/s at 35o
above the horizontal. How long is it in air?

1. 6.69 s
2. 8.16 s
3. 2.34 s
4. 4.08 s
5. 4.68 s

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A ball is kicked straight up from ground level
with initial velocity of 22.6 m/s. How high above
the ground will the ball rise?

1. 9.8 m
2. 3.00 m
3. 1.15 m
4. 26.1 m
5. 19.6 m

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Summary of Concepts

• X and Y directions are Independent!
• Position, velocity and acceleration are vectors
• F m a applies in both x and y direction
• Projective Motion
• ax 0 in horizontal direction
• ay g in vertical direction

50
60
1. A car initially traveling at a velocity vo
begins to slow down with a uniform deceleration
of 1.20 m/s2 and comes to a stop in 26.0 seconds.
Determine the value of vo.

1. 31.2 m/s
2. 21.7 m/s
3. 27.2 m/s
4. 24.8 m/s
5. None of these

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2. A 102.0-kg person stands on a scale inside an
elevator moving downward with anacceleration of
1.300 m/s2. What will be his apparent weight?

1. 999.6 N
2. 132.6 N
3. 867.0 N
4. 1132 N
5. 0 N

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3. A ball is thrown with a speed of 27.0 m/s at
35o above the horizontal. At the maximum height,
its speed will be

1. 27.0 m/s
2. -9.8 m/s
3. 0 m/s
4. 22.1 m/s
5. 15.5 m/s

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4. A ball is thrown with a speed of 32.0 m/s at
50o above the horizontal. How long is it in air?

1. 4.20 s
2. 6.53 s
3. 3.27 s
4. 2.50 s
5. 5.00 s

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5. A ball is kicked straight up from ground level
with initial velocity of 16.6 m/s. How high above
the ground will the ball rise?

1. 28.1 m
2. 14.1 m
3. 1.69 m
4. 0.847 m
5. 1.18 m

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6. A block is at rest on a flat board. The flat
board is then gently tilted. If the block starts
to slide at a tilt angle of 23.8o, what is the
coefficient of static friction (?s) between the
block and the board?

1. 0.40
2. 0.91
3. 23.8
4. 87.6
5. 0.44

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