Download Horizontal Motion

Projectile Motion
What is a Projectile?
• Anything that is thrown or shot through
the air.
• Projectiles have velocities in two
directions.
– Horizontal Motion: Motion parallel to the
Earth’s surface.
– Vertical Motion: The force of gravity
pulling down on the object.
Projectile Motion
• A projectile’s horizontal and vertical
motion are completely independent of
each other.
• Gravity will effect a projectile and a falling
object in the same way.
• Therefore, if an object is dropped and
thrown at the same time they will hit the
ground at the same time. It does not
matter that the projectile travels a farther
distance.
Newton’s Cannon
Newton
described the possible
trajectories of a cannonball shot
from a tall mountain. Weaker shots
fall in parabolas, but soon the
curvature of the earth becomes
more important. If you could ignore
air resistance, stronger shots
would clear the horizon. If you
could launch the cannonball fast
enough, its curve would match the
curvature of the earth. This is
called an orbit. A satellite is no
more than a projectile moving fast
enough to continually clear the
horizon as it falls.
Curved Path
• A projectile is any object that is projected
by some means and continues in motion
by its own inertia. Without gravity, the
object would follow a linear motion; with
gravity, the path curves.
• The path is surprisingly simple if the
horizontal and vertical components of
motion are investigated separately.
HORIZONTAL COMPONENT:
• If friction is ignored (and we always ignore
friction), then there are no forces acting on
the x-axis.
• If net force is zero, then acceleration is zero.
• If acceleration is zero, then the object is either at
rest or moving at constant velocity.
• Since we know that the object is already in motion,
the projectile must be moving at constant speed on
the x-axis.
• It moves of its own inertia and covers equal
distances in equal intervals of time.
VERTICAL COMPONENT:
• Projectiles move just like a freely falling
object along the y-axis.
• The changing motion is due only to
acceleration due to gravity.
– On the way up, the object decreases its
speed as it goes against gravity.
– On the way down, speed increases as it
moves with gravity.
Projection Angles
• With the same initial speed but different projection angles, a
projectile will reach different altitudes (height above the ground)
and different ranges (distances traveled horizontally).
• However, the same range can be obtained from two different
angles, symmetrically around a maximum of 45˚, as shown in the
graph.
Symmetry
• The path of a projectile is symmetrical…
• It rises to its maximum height in the same time it
takes to fall from that height to the ground.
• Because acceleration is the same all of the time,
the speed it loses while going up is the same as
the speed it gains while falling.
• Therefore the speeds are the same at equal
distances from the maximum height, where the
vertical speed is zero.
WHAT IF THE PROJECTILE
IS SHOT UPWARD?
Now assume gravity is turned on!
The projectile would travel
with a parabolic trajectory.
The downward force of
gravity will act upon the
cannonball to cause the
same vertical motion as
before - a downward
acceleration. The cannonball
falls the same amount of
distance in every second as
it did when it was merely
dropped from rest.
REVIEW TO THIS POINT!
• a projectile is any object upon which the only force acting upon
it is gravity,
• projectiles travel with a parabolic trajectory due to the influence
of gravity,
• there are no horizontal forces acting upon projectiles and thus
no horizontal acceleration,
• the horizontal velocity of a projectile is constant (never
changing in value),
• there is a vertical acceleration caused by gravity; its value is
• -9.8 m/s/s, down,
• the vertical velocity of a projectile changes by 9.8 m/s each
second,
• the horizontal motion of a projectile is independent of its
vertical motion.
PROJECTILE MOTION
EQUATIONS
• The following are the formulas you will need to solve
projectile motion problems. They are derived from
the kinematic equations.
y = Vyt + 1/2gt2
x = Vxt
Vy = Vyo + gt
Example
(Horizontally Launched Projectile):
A steel ball rolls with a constant
velocity across a table top 0.598m high.
The steel ball rolls towards the edge
and falls to the ground. When it
eventually hits the ground, it had
travelled 0.852m horizontally from the
edge of the table. How fast was the ball
rolling?
Solution:
Given:
x = 0.852 m
Y = - 0.598 m
Vyo = 0 m/s
Equations:
Unknown: Vx = ?
x = V xt
y = vyt + ½ gt2
Vy = Vyo + gt
Solution (Cont’d):
y = Vyt + ½ gt2
y = ½ gt2 (Vy = 0 m/s)
t2 = 2y/g
t = √(2y/g)
t = √[(2)(-0.598 m) / (-9.8 m/s2)
t = 0.34 s
Solution (Cont’d):
Therefore:
x = Vxt
Vx = x/t
Vx = 0.852 m / 0.34 s
Vx = 2.5 m/s
LAUNCHED AT AN ANGLE
• What if we have an object that is launched
at an angle and lands at the same height
as it was launched?
EXAMPLE PROBLEM
Bubba Watson crushes another drive off
the tee at The Masters golf tournament.
He drives the ball at a 30° angle with a
velocity of 38 m/s.
a.) How long does it take for the golf ball to
reach its maximum height?
b.) What is the maximum height?
c.) What is the length of the drive?
SOLUTION