Projectile Motion
Notes and Explanations
Some diagrams and simulations from
www.physicsclassroom.com
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.
Simulation of Newton’s Cannon
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 xaxis.
• 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.
Vector Combination
• The curved path of a projectile is a
combination of these horizontal
and vertical components. The
curved path of a ball (as shown in
the diagram) is best analyzed by
considering the horizontal and
vertical components of motion
separately.
• Note that the ball’s horizontal
component of motion doesn’t
change as the falling ball moves
forward.
• Note the vertical positions become
farther apart with time and that the
vertical distances traveled are the
same as if the ball were simply
dropped.
Projection Angles
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•
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.
Questions:
• At the instant a horizontally held rifle is fired over level ground, a
bullet held at the side of the rifle is released and drops to the
ground. Ignoring air resistance, which bullet strikes the ground first?
• A projectile is launched at an angle into the air. If air resistance is
negligible, what is the acceleration of its vertical component of
motion? Of its horizontal component of motion?
• At what part of its trajectory does a projectile have minimum speed?
• A ball tossed into the air undergoes acceleration while it follows a
parabolic path. When the sun is directly overhead, does the shadow
of the ball across the field also accelerate?
• Draw a free body diagram of a projectile.
• What type of motion occurs on the horizontal axis?
• How in the world can object be moving upward if the only force
acting upon it is gravity?
• Are horizontal forces required to keep the projectile moving
horizontally?
Answers
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They strike at the same time.
-9.8 m/s2; 0
At maximum height.
NO!
Should show weight only.
Constant
Inertia
NO!
Memorize the following conceptual
notions about projectiles:
• a projectile is any object upon which the only force 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 ~10
m/s2, down,
• the vertical velocity of a projectile changes by ~10 m/s each second,
• the horizontal motion of a projectile is independent of its vertical
motion.
• the speed of a projectile is symmetrical around its maximum height.
Use these sites to answer the
questions on the next slide.
Simulation #1
Simulation #3
Simulation #2
Simulation #4
Questions
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What launch angle maximizes the range (horizontal distance)?
What launch angle maximizes the height reached?
How does the time spent in the air depend on the launch angle?
Compare what happens at one angle, and then at 90 degrees minus
that angle (30 and 60, for instance).
What does Newton’s Cannon show?
What happens to the trajectory when the mass of the projectile is
changed?
What effect does air resistance have on a trajectory?
What happens to the trajectory when the initial velocity is changed?
What conditions must exist so that two projectiles will collide in
midair? What effect do the initial velocities, height differences, and
angles have on this collision?
Compare the trajectories of a projectile that is under the influence of
gravity and one that is not?