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Material Type: Notes; Professor: Benton; Class: GENERAL PHYSICS; Subject: Physics ; University: Oklahoma State University - Stillwater; Term: Fall 2014;
Typology: Study notes
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Fall 2009Oklahoma State University
PHYS2014: BentonLecture 4, Slide 1
θ
G v^^0
G h
If I shoot an arrow into the air with initial velocity
and at angle
, it follows a parabolic trajectory. The horizontal (
x
) distance it
travels is called the range,
. The maximum vertical (
y
) distance
it reaches half way through its flight we’ll call the height,
v
G h
PHYS2014: BentonLecture 4, Slide 2
As in the inclined track example, we must treat the
x
and
y
components of motion separately (two 1-D problemsinstead of one 2-D problem).The initial velocity in the
x
direction will be
The initial velocity in the
y
direction will be
0
0
cos
x v
v
0
0
sin
y v
v
θ
0
sin 0
= G^
G
v^ y
v
θ
0
cos 0
= G^
G
v^ x
v
θ
G h
PHYS2014: BentonLecture 4, Slide 4
(^0) The arrow takes the same time to fly from its maximumheight back to the ground as it takes to reach itsmaximum height from the ground. The total time offlight is then:
10
20
30
40
50
60
70
80
90
100
Range (m)
(^25201510) Height (m)^50
0 2
sin
v
t^
g
0
1
2
3
4
5
Time (s)
(^25201510) Height (m)^50
Fall 2009Oklahoma State University
PHYS2014: BentonLecture 4, Slide 5
Since there is no acceleration in the
x
direction, the
horizontal velocity of the arrow is constant.
0
cos
x v
v
The range,
, then will just be the time of flight,
, times
the horizontal velocity,
0
0
0
2 2
sin
cos
sin
cos
sin
cos
o
x
v o
v v
v T
v
g
g
v
i
g
Apply the trignometric identity
sin
cos
sin 2 2
θ
θ =
θ
and the total range will be:
G vx
sin 2
o v
i
g
Fall 2009Oklahoma State University
PHYS2014: BentonLecture 4, Slide 7
What angle leads to the maximum height?Height will be maximum when all of
is in the vertical (
y
direction and there is minimum (zero) horizontal (
x
component.Not surprisingly, the maximum height is reached by shootingthe arrow straight up.
v
PHYS2014: BentonLecture 4, Slide 8
0
10
20
30
40
50
60
70
80
90
100
Range (m)
(^3530252015) Height (m) 10 5 0
30 deg.45 deg.60 deg.
0
1
2
3
4
5
6
Time (s)
(^3530252015) Height (m) 10 5 0
30 deg.45 deg.60 deg.
Projectile Motion using different angles,
, and
0
G v
PHYS2014: BentonLecture 4, Slide 10
30 m s,
v
a
30 m s,
v
a
PHYS2014: BentonLecture 4, Slide 11
Because coffee is a liquid, it is not rigidly fixed to thecup.On the other hand, the cup is fixed rigidly to my hand,my hand is fixed rigidly to the rest of my body, and mybody is fixed rigidly to the car by virtue of the fact I’msitting in the passenger’s seat with my seat belt on.The coffee has
mass
and one property of mass is
inertia
Inertia
is the tendency of an object to resist a change in
its motion (i.e. resist a change in its velocity).When the car suddenly accelerates, the inertia of thecoffee causes it to want to remain at rest and so itsloshes backwards in the cup.
PHYS2014: BentonLecture 4, Slide 13
st
I shoot an arrow into the air with initial velocity
The arrow follows a parabolic trajectory which is the two-dimensional sum of its (1-D) horizontal (
x
) motion and its
(1-D) vertical (
y
) motion.
v
θ
0
sin 0
= G^
G
v^ y
v
θ
0
cos 0
= G^
G
v^ x
v
θ
G h
PHYS2014: BentonLecture 4, Slide 14
θ
G F^ g
G F^ g
G h
If we consider just the vertical component of motion, weknow that the arrow will reach some maximum height andwill then start falling back towards the ground. Weexplained that this is due to the acceleration of
gravity
Gravity is a
force
acting on the arrow in the
y
direction and
and continually affecting its motion (i.e. its velocity).
Fall 2009Oklahoma State University
PHYS2014: BentonLecture 4, Slide 16
0
10
20
30
40
50
60
70
80
90
100
Range (m)
(^25201510) Height (m)^50
0
1
2
3
4
5
Time (s)
(^25201510) Height (m)^50
0
1
2
3
4
5
Time (s)
9080706050 403020100 100 Height (m)
Vertical Motion: projectile isacted on by force of gravity
Horizontal Motion: no forceis acting on the projectile
Net sum of the vertical andhorizontal components ofmotion: a parabolic trajectory
Fall 2009Oklahoma State University
PHYS2014: BentonLecture 4, Slide 17
While the airplane is in level flight and at constantvelocity, there are no horizontal forces acting on thebowling ball. The bowling ball doesn’t move relative tothe plane. The airplane is anAt take off, the plane is accelerating forward. Thebowling ball rolls toward the back of the plane. Theairplane is in a non-inertial frame of reference.
inertial frame of reference.
Bowling Ball on Airplane Floor
PHYS2014: BentonLecture 4, Slide 19
“A force is any influence that can produce a change in thevelocity of an object.”
a rather circular definition.
Newton’s 2
nd
Law of Motion:
or in words“The net force acting upon an object is equal to theproduct of the mass and the acceleration of the object; thedirection of the force is the same as the object’sacceleration.”
G
PHYS2014: BentonLecture 4, Slide 20
gravity
When gravity acts on an object, it causes the object tofall.
fall with a constant acceleration of 9.80 m/s
will cause that object to
accelerate
, i.e. change its
velocity Q. What determines the magnitude of the force of gravity
on an object? A. The object’s mass. (Remember, inertia is a property of
mass).
