Transcript Lecture 5.Projectile..

Projectile Motion
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
Both boys jump off the cliff into the water at the
same time. Boy 2 has an additional horizontal
velocity. Which boy lands in the water first?
A) Boy 1.
B) Boy 2.
C) They land at the
same time.
D) Cannot tell
without further
information.
A
B
Answer: C
The horizontal and vertical motion is independent
of each other. Both start with zero velocity in the
y-direction and have the same acceleration. Their
vertical motion is identical.
Last Time
Vectors
Today
Two & three dimensional motion
Projectile motion
Relative motion
r is a position
vector from the
origin to a given
point.
r = r2  r1
displacement vector
Displacement vectors do not depend on origin
of coordinate system! See next slide.
r is a position
vector from the
origin to a given
point.
r1
r2
r = r2  r1
displacement vector
Displacement vectors do not depend on origin
of coordinate system!
Position vector = r
(drawn from origin to point)
Displacement vector = r  rf  ri
r
Average velocity vector = vav 
t
dr
Instantaneous velocity vector = v =
dt
v
Average acceleration vector = aav 
t
dv
Instantaneous acceleration vector = a 
dt
Velocity and Acceleration Vectors for a
Particle Moving Along a Winding Path
Trajectory
r
r
Constant-Acceleration
Equations of Motion
Position as a
function of
time
Velocity as a
function of
time
Velocity as a
function of
position
x = x0 + v0xt
+ ½ axt2
vx = v0x
+ axt
vx2 = v0x2
+ 2axx
y = y0 + v0yt
+ ½ ayt2
vy = v0y
+ ayt
vy2 = v0y2
+ 2ayx
Two dimensional motion
Horizontal and vertical motions are
independent! It is that simple.
We will find later that we only need
to look at the force components along
each direction. It is force that causes
acceleration.
Velocity, Addition from Position
Vector. Suppose the position of an
object is given by
r  (3.0t i  6.0t j )m
2
3
(a) Determine its velocity and
acceleration as a function of time.
(b) Determine r and v at time
t  2.5 s .
Projectile motion
Assumptions:
 air resistance is ignored for now.
 acceleration of gravity is constant
and has value g = 9.80 m/s2.
 Earth’s rotation is ignored.
A projectile is an
object moving in two
dimensions under the
influence of Earth's
gravity; its path is a
parabola.
Water fountains show projectile
motion
Set up solutions
Choose coordinate
system.
y
ay =  g
x
ax = 0 and ay =  g
Now what will the earlier kinematic
equations look like?
Projectile motion equations
x  x0  v0 xt
1 2
y  y0  v0 y t  gt
2
vx  v0 x
v y  v0 y  gt
v v
2
x
2
0x
v  v  2 g y
2
y
2
0y
Don’t
memorize these
equations!
Conceptual Quiz: Look at the demo
(shoot and drop). Two balls are released
at the same time by a spring loaded
mechanism. Ball A falls straight down.
Ball B is propelled out horizontally.
Which ball lands first?
A)
B)
C)
D)
Ball A
Ball B
Impossible to determine.
Balls land at the same time.
Answer
D) Balls land at the same time.
Both balls feel the same vertical
acceleration, -g. Therefore, they drop
at the same rate.
Of course their horizontal motion is
quite different.
Do demo.
It can be
understood by
analyzing the
horizontal and
vertical motions
separately.
The speed in the x-direction
is constant; in the ydirection the object moves
with constant acceleration g.
This photograph shows two balls
that start to fall at the same time.
The one on the right has an initial
speed in the x-direction. It can be
seen that vertical positions of the
two balls are identical at identical
times, while the horizontal
position of the yellow ball
increases linearly.
Both boys leave at the same time and
land at the same time. What is d?
h
d
Special Extra 3. Work out solution
to previous slide.
Problem solution video.
Demo:
Projectile motion (monkey and
the hunter)
If an object is launched at an initial angle of θ0 with the
horizontal, the analysis is similar to boy jumping off cliff
except that the initial velocity has a vertical component.
Notice speed changes continuously. Surprise demo.
Consider equations for projectile motion as in
lab manual. Range, time, etc. These equations
are worked out in most textbooks. Work these
out yourself. T is time of flight.
2
v0 sin 2q
R=
g
2v0 sin q
T=
g
2
0
2
v sin q
h=
2g
h
Conceptual Quiz
A projectile is
launched from the
ground at an angle of
30°. At what point in
its trajectory does
this projectile have
the least speed?
A) just after it is launched
B) at the highest point in its
flight
C) just before it hits the
ground
D) halfway between the
ground and the highest
point
E) speed is always constant
Conceptual Quiz
A projectile is launched
from the ground at an
angle of 30º. At what
point in its trajectory
does this projectile have
the least speed?
The speed is smallest
at the highest point of
its flight path because
the
y-component of the
velocity is zero.
A) just after it is launched
B) at the highest point in its
flight
C) just before it hits the ground
D) halfway between the ground
and the highest point
E) speed is always constant
Conceptual Quiz
Which of
the three
punts has
the longest
hang time?
h
A
B
D) all have the same hang time
C
Conceptual Quiz
Which of
the three
punts has
the longest
hang time?
h
A
B
D) all have the same hang time
The time in the air is determined by the
vertical motion! Because all of the punts reach
the same height, they all stay in the air for the
same time.
C
Conceptual Quiz
h
A
B
C
D) all have the same
Follow-up: which one had the greater initial velocity?
Conceptual Quiz
h
A
B
D) all have the same
Follow-up: which one had the greater initial velocity?
C
Conceptual Quiz
A battleship simultaneously fires two shells at
enemy ships. If the shells follow the parabolic
trajectories shown, which ship gets hit first?
A) Ship A.
B) Ship B.
C) Both at the same time.
D) Need more information.
The answer is B.
Consider the time for the shell to reach
its maximum height (or fall from its
maximum height). Since shell A goes
higher, it takes a longer time than shell B.
Projectile Motion Applet
http://physics.bu.edu/~duffy/semester
1/semester1.html
Questions: Three Projectiles
Maximum height
Time of flight
Range
Monkey and the Hunter
http://galileo.phys.virginia.edu/classes
/109N/more_stuff/Applets/Projectile
Motion/jarapplet.html
Projectiles with Air Resistance
Relative Velocity in Two Dimensions
vtg  velocity of train relative to ground
vpt  velocity of person relative to train
vpg  vpt +vtg = velocity of person relative to ground
v13  v12  v23
Relative speed in one dimension is
similar in two dimensions except that
we must add and subtract velocities as
vectors.
Each velocity is labeled first with the
object, and second with the reference
frame in which it has this velocity.
Here, vWS is the velocity of
the water in the shore
frame, vBS is the velocity of
the boat in the shore frame,
and vBW is the velocity of
the boat in the water frame.
The relationship between
the three velocities is:
Discuss relativity car demo.
Do not do it before conceptual quiz.
Conceptual Quiz
A small cart is rolling
at constant velocity on
a flat track. It fires a
ball straight up into the
air as it moves. After
it is fired, what
happens to the ball?
A) it depends on how fast the
cart is moving
B) it falls behind the cart
C) it falls in front of the cart
D) it falls right back into the
cart
E) it remains at rest
Conceptual Quiz Answer
A small cart is rolling
at constant velocity on
a flat track. It fires a
ball straight up into the
air as it moves. After
it is fired, what
happens to the ball?
In the frame of reference of
the cart, the ball only has a
vertical component of
velocity. So it goes up and
comes back down. To a
ground observer, both the
cart and the ball have the
same horizontal velocity,
so the ball still returns into
the cart.
A) it depends on how fast the
cart is moving
B) it falls behind the cart
C) it falls in front of the cart
D) it falls right back into the
cart
E) it remains at rest
when
viewed from
train
when
viewed from
ground
Airplane Drop. The pilot of an
airplane traveling 170 km/h wants to
drop supplies to flood victims
isolated on a patch of land 150 m
below. The supplies should be
dropped how many seconds before
the plane is directly overhead?
Projectile Angle. At what
projection angle will the range of
a projectile equal its maximum
height?
Solving Problems Involving
Projectile Motion
1. Read the problem carefully, and choose the
object(s) you are going to analyze.
2. Draw a diagram.
3. Choose an origin and a coordinate system.
4. Decide on the time interval; this is the same in
both directions, and includes only the time the
object is moving with constant acceleration g.
5. Examine the x and y motions separately.
Copyright © 2009 Pearson Education, Inc.
Solving Problems Involving
Projectile Motion
6. List known and unknown quantities.
Remember that vx never changes, and
that vy = 0 at the highest point.
7. Strategy. Plan how you will proceed. Use
the appropriate equations; you may have
to combine some of them.
8. Find the solution.
Copyright © 2009 Pearson Education, Inc.