Transcript Two-Dimensional Motion and Vectors Chapter 3

Two-Dimensional Motion
and Vectors
CP: 6.1
Projectile Motion Assumptions
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The acceleration of gravity is a constant
-9.8 m/s2
The effect of air resistance is
negligible
The rotation of the Earth has no effect.
Projectile motion only applies
to bodies in free fall
Not in free
fall
Projectiles are moving in 2
dimensions
Therefore, we need to look in two
dimensions (the x-direction & ydirection) when solving projectile
problems.
The motion on the y axis is independent of the
motion on the x axis.
y axis
free fall motion
x axis
constant velocity motion.
We will see in the next chapter,
this is Newton’s First Law of Motion.
This leads to a parabolic path
On the horizontal
Dx = v  t
On the vertical
Dx = vit+½ at2
For Example…
A cannon has a muzzle
velocity of 62.3 m/s. What is its
range when shot at an angle of
30.00o?
Example: A cannon has a muzzle velocity of
62.3 m/sec. What is its range when shot at an
angle of 30.00o?
1. Draw a vector diagram, and resolve the
velocity vector into rectangular
components.
62.3 m/s
30o
62.3cos30
Range ( Dx)
2. Using the y axis component, and the equations of
motion for free fall, calculate the time of flight. (How
long the projectile is in the air)
(Y axis motion only)
vi = 62.3sin30 = 31.15 m/sec
a = -9.8 m/sec2
62.3sin30
Dy = 0
t=?
Dy = vit + ½ at2
0 = 31.15t + ½(-9.8)t2
0 = (t)(31.15 – 4.9t)
t = 6.357sec
3. Using the time of flight, calculate how
far the projectile will travel horizontally
during that time.
X Axis Motion Only
Dx = vx t
Dx = 62.3 cos30 x 6.357 sec
Dx = 53.95 m/sec x 6.357 sec
Dx = 342.96 ~ 343 m
The maximum
range of a
projectile
occurs at 45o.
Misconception #1
Going faster horizontally means you
don’t fall as fast.
Misconception #2:
Gravity won’t act on you until you look
down.
That is just
so wrong!
A battleship simultaneously fires two shells at
enemy ships. If the shells follow the trajectories
shown, which ship gets hit first?
A
B
1. A will hit first
3. Both will hit at the same time
2. B will hit first
4. Depends on the actual angles.
Initial velocity
vector
Example #2
75 ft
60o
R ft
A golfer makes a shot to a tee as
shown. Assuming he shoots at a 60.0o
angle, with a velocity of 100. ft/sec what is
the range (dx) to the tee? (UNITS!)
100 sin 60
100
75 ft
60o
100 cos 60
Find components of the initial
velocity vector
Using our standard equations of
motion…
On the y axis
a = -32 ft/sec2
viy = 100 sin 60o
d = + 75 ft
t=?
Vertical displacement
when the ball is at
the elevation of the
tee
On the y axis
a = -32 ft/sec2
viy = 100 sin 60o
= 86.6
d = + 75 ft
t=?
d = vit + ½ at2
75 = (86.6)t + (-16)t2
-16t2 + 86.6t – 75 = 0
T = 1.08 sec. & 4.33sec
As per the diagram,
assume the long shot.
4.33 sec
1.08
sec
75ft
60o
R ft
On the x axis…
v = 100cos60o = 50 ft/sec
t = 4.33 sec
Range ( R) = vx  t
= 50 ft/sec (4.33 sec) = 217 ft
Which ball spends more time in the air?
Which ball has the greater launch speed?
same
B
The time of flight depends only on
the vertical component of the initial
velocity. In this case, the vertical
component is the same, ie—both balls
reached the same height, so they will
spend the same time in the air.
Since Ball A has the shorter range,
the horizontal component of its initial
velocity must be less than that of Ball B.
So Ball A has a smaller launching speed.
Which ball spends more time in the air?
Which ball has the greater launch speed?
Ball B spends more
time in the air.
Ball B has the
greater launch
speed.
Ball B spends more time in the air.
Again, the time of
flight depends only on
the vertical component
of the initial velocity.
Ball B goes higher, so
it must spend more
time in the air.
Ball B has the greater launch speed.
Both balls have the
same range. We know
that 45o gives maximum
range for a given speed.
Equivalently, 45o is the
angle required for the
smallest launch speed to
achieve a given range.
Ball B has the greater launch speed.
The closer the launch
angle is to 45o, the
closer the launch speed
is to this smallest
speed. The launching
angles of both balls is
greater than 45o. But,
notice that Ball A’s
launch angle is closer to
45o than Ball B’s. So
Ball A has the smaller
launch speed of the
two.