Acknowledgement
I thank my dear teachers to give such a wonderful topic and my
institution for giving me such wonderful opportunities.
I would also like to thank my parents for supporting me throughout
the project.
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Sr no.
IndexPage No.
1. Introduction 3
2. History of Projectile Motion 5
3. Horizontally Launched Projectiles 8
4. Non-Horizontally Launched 10
Projectiles
5. Applications 13
6. Our Investigation 17
7. Conclusion 19
8. Bibliography 20
9. Remarks 22
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Introduction
A projectile is an object upon which the only force acting is
gravity. Projectile is any object that once projected or dropped
continues in motion by its own inertia and is influenced only by the
downward force of gravity. By definition, a projectile has a single
force that acts upon it - the force of gravity. If there were any other
force acting upon an object, then that object would not be a projectile.
Thus, the free-body diagram of a projectile would show a single force
acting downwards and labelled force of gravity (or simply Fgrav). By
definition, a projectile is any object upon which the only force is
gravity.
Projectile Motion and Inertia
Many students have difficulty with the concept that the only
force acting upon an upward moving projectile is gravity. Their
conception of motion prompts them to think that if an object is
moving upward, then there must be an upward force. And if an object
is moving upward and rightward, there must be both an upward and
rightward force. Their belief is that forces cause motion; and if there
is an upward motion then there must be an upward force. They
reason, "How in the world can an object be moving upward if the only
force acting upon it is gravity?" Newton's laws suggest that forces are
only required to cause an acceleration (not a motion). The Newton's
laws stood in direct opposition to the common misconception that
a .force is required to keep an object in motion. This idea is
simply not true! A force is not required to keep an object in motion. A
force is only required to maintain an acceleration. And in the case of a
projectile that is moving upward, there is a downward force and a
downward acceleration. That is, the object is moving upward and
slowing down.
To further ponder this concept of the downward force and a
downward acceleration for a projectile, consider a cannonball shot
horizontally from a very high cliff at a high speed. And suppose for a
moment that the gravity switch could be turned off such that the
cannonball would travel in the absence of gravity? What would the
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motion of such a cannonball be like? How could its motion be
described? According to Newton's first law of motion, such a
cannonball would continue in motion in a straight line at constant
speed. If not acted upon by an unbalanced force, "an object in motion
will continue to remain at rest or in motion until acted by an external
force"
Now suppose that the gravity switch is turned on and that the
cannonball is projected horizontally from the top of the same cliff.
What effect will gravity have upon the motion of the cannonball? Will
gravity affect the cannonball's horizontal motion? Will the cannonball
travel a greater (or shorter) horizontal distance due to the influence of
gravity? The answer to both of these questions is "No!" Gravity will
act downwards upon the cannonball to affect its vertical motion.
Gravity causes a vertical acceleration. The ball will drop vertically
below its otherwise straight-line, inertial path. Gravity is the
downward force upon a projectile that influences its vertical motion
and causes the parabolic trajectory that is characteristic of projectiles.
A projectile is an object upon which the only force is gravity.
Gravity acts to influence the vertical motion of the projectile, thus
causing a vertical acceleration. The horizontal motion of the projectile
is the result of the tendency of any object in motion to remain in
motion at constant velocity. Due to the absence of horizontal forces, a
projectile remains in motion with a constant horizontal velocity.
Horizontal forces are not required to keep a projectile moving
horizontally. The only force acting upon a projectile is gravity!
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History on Projectile Motion
It was Galileo who first accurately described projectile motion.
He showed that it could be understood by analysing the horizontal
and vertical components separately. No one had done this prior to
Galileo.
This illustration reflects the
general opinion before Galileo
which followed largely
Aristotelian lines but
incorporating a later theory of
"impetus" -- which maintained
that an object shot from a
cannon, for example, followed
a straight line until it "lost its
impetus," at which point it fell
abruptly to the ground.
Later, simply by more careful
observation, as this illustration
from a work by Niccolo
Tartaglia shows, it was realized
that projectiles actually follow
a curved path. Yet no one
knew what that path was, until
Galileo. There was yet another
brilliant insight that led Galileo
to his most astounding
conclusion about projectile
motion. First of all, he
reasoned that a projectile is not
only influenced by one motion,
but by two. The motion that
acts vertically is the force of
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gravity, and this pulls an object
towards the earth at 9.8 meters
per second. But while gravity
is pulling the object down, the
projectile is also moving
forward, horizontally at the
same time. And this horizontal
motion is uniform and constant
according to Galileo's principle
of inertia. He was indeed able
to show that a projectile is
controlled by two independent
motions, and these work
together to create a precise
mathematical curve. He
actually found that the curve
has an exact mathematical
shape. A shape that the Greeks
had already studied and called
the parabola. The conclusion
that Galileo reached was that
the path of any projectile is a
parabola.
At a given location on the earth and in the absence
of air resistance, all objects fall with the same
uniform acceleration. Thus, two objects of different
sizes and weights, dropped from the same height,
will hit the ground at the same time.
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An object is controlled by two independant
motions. So an object projected
horizontally will reach the ground in the
same time as an object dropped vertically.
No matter how large the horizontal
velocity is, the downward pull of gravity is
always the same.
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Horizontally Launched Projectiles
Consider a cannonball projected horizontally by a cannon from
the top of a very high cliff. In the absence of gravity, the cannonball
would continue its horizontal motion at a constant velocity. This is
consistent with the law of inertia. And furthermore, if merely dropped
from rest in the presence of gravity, the cannonball would accelerate
downward, gaining speed at a rate of 9.8 m/s every second. This is
consistent with our conception of free-falling objects accelerating at a
rate known as the acceleration of gravity.
If our thought experiment continues and we project the
cannonball horizontally in the presence of gravity, then the
cannonball would maintain the same horizontal motion as before - a
constant horizontal velocity. Furthermore, the 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 as it did when it was merely dropped from rest (refer to
diagram below). However, the presence of gravity does not affect the
horizontal motion of the projectile. The force of gravity acts
downward and is unable to alter the horizontal motion. There must be
a horizontal force to cause a horizontal acceleration. (And we know
that there is only a vertical force acting upon projectiles.) The vertical
force acts perpendicular to the horizontal motion and will not affect it
since perpendicular components of motion are independent of each
other. Thus, the projectile travels with a constant horizontal
velocity and a downward vertical acceleration.
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Equation:-
1) Equation of Trajectory:
a) Horizontal Equation=x=ut
b) Vertical Equation=y=1/2gt2
This is the equation of a parabola, symmetric about y-axis.
Hence the path of the projectile is parabolic.
2) Time Of Descent: Time taken for the projectile to come to the
earth.
t= √ g
2h
3) Horizontal Range=X=distance travelled by the projectile in
horizontal direction
X=u* √ g
2h
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Non horizontal projectile motion
Now suppose that our cannon is aimed upward and shot at an
angle to the horizontal from the same cliff. In the absence of gravity
(i.e., supposing that the gravity switch could beturned off) the
projectile would again travel along a straight-line, inertial path. An
object in motion would continue in motion at a constant speed in the
same direction if there is no unbalanced force. This is the case for an
object moving through space in the absence of gravity. However, if
the gravity switch could be turned on such that the cannonball is truly
a projectile, then the object would once more free-fall below this
straight-line, inertial path. In fact, the projectile would travel with
a parabolic trajectory. The downward force of gravity would 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 (refer to diagram below). Once more, the presence of gravity does
not affect the horizontal motion of the projectile. The projectile still
moves the same horizontal distance in each second of travel as it did
when the gravity switch was turned off. The force of gravity is a
vertical force and does not affect horizontal motion; perpendicular
components of motion are independent of each other.
In conclusion, projectiles travel with a parabolic trajectory due
to the fact that the downward force of gravity accelerates them
downward from their otherwise straight-line, gravity-free trajectory.
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This downward force and acceleration results in a downward
displacement from the position that the object would be if there were
no gravity. The force of gravity does not affect the horizontal
component of motion; a projectile maintains a constant horizontal
velocity since there are no horizontal forces acting upon it.
Projectile Motion Formula (trajectory formula) is given by
Where Vx is the velocity along x-axis,
Vxo is the initial velocity along x-axis,
Vy is the velocity along y-axis,
Vyo is the initial velocity along y-axis.
g is the acceleration due to gravity and
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t is the time taken.
Equations related to trajectory motion (projectile motion) are given by
Where Vo is the initial Velocity,
sin θ is the component along y-axis,
cos θ is the component along x-axis.
Projectile Motion formula is used to find the distance, velocity
and time taken in the projectile motion.
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Applications of Projectile Motion
1) In Sports:
Cricket
Javelin throw
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Rugby
Football
2) Powered Projectiles, Rockets And Missiles and Canons:
Advanced military techniques include missile launching which
uses projectile motion equations to calculate the distance at where
the missile is going to fall. The same goes for canons, catapults,
ballista, trebuchets and rockets.
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Missile:-
Canons
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Trebuchet:-
Fountains: - Projectile motion concept is used to determine
the height of the fountain of water and the distance it will go.
It’s used in building fountains.
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Our Investigation:
Aim: - To investigate the velocity and the
maximum height reached by a projectile
fired at an angle made with the horizontal
from the range and angle θ and the time
period from the made model.
Apparatus: -
Observation Table: -
Angle Range Velocity Height Time
Θ Rm V m/s Hm Ts
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30 0.81 3.027 0.116 0.30
40 1.18 3.426 0.247 0.45
50 0.82 2.856 0.244 0.44
60 0.8 3.008 0.346 0.53
70 0.63 3.099 0.432 0.59
Angl Sin Rg Sin2Θ Sq. Root of Sq. Root of
e Θ Sin2Θ Rg
30 0.5 29 0.866 9.31E-01 9.65E-01
4
40 0.64 39 0.984 9.92E-01 9.96E-01
2 2
50 0.76 49 0.984 9.92E-01 9.96E-01
6 0
60 0.86 58 0.866 9.31E-01 9.65E-01
6 8
70 0.93 68 0.642 8.01E-01 8.95E-01
9 6
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Result: - As the angle θ increases, the range of
the projectile increases till 450 and then decreases
till 900. The velocity component remains the
same as we are stretching the rubber the same
way every time
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Conclusion
s
Projectile is a very interesting topic of itself. It shows us the
beauty of physics in a completely different way. It helps us calculate
basic components of parabolic motion such as velocity, range and
maximum height etc without the consideration of mass of the body
which makes it less complicate to calculate the components.
Before this investigatory projectile was just the part of the
physics book but now after knowing the applications of projectile our
concepts about it are clear. By doing this project we learned to
connect projectile motion to many real life situations.
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Bibliography
http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Projectile-
Motion.html
http://www.physicsclassroom.com/mmedia/vectors/hlp.cfm
http://www.physicsclassroom.com/class/vectors/u3l2e.cfm
http://serc.carleton.edu/sp/compadre/uncertainty/examples/example5.html
http://answers.yahoo.com/question/index?qid=20070519032007AAcE7UV
http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3-Physics-Vol-1/Projectile-
Motion-Real-life-applications.html
http://homepage.usask.ca/~dln136/projectile/pages/module5.html
http://homepage.usask.ca/~dln136/projectile/pages/module3.html
http://formulas.tutorvista.com/physics/projectile-motion-formula.html
http://www.physicsclassroom.com/class/vectors/u3l2a.cfm
http://en.wikipedia.org/wiki/Projectile_motion
http://www.physicsclassroom.com/Class/vectors/U3L2b.cfm
http://www.physicsclassroom.com/Class/vectors/U3L2c2.cfm
http://www.physicsclassroom.com/class/vectors/U3L2e.cfm
http://usnavymuseum.org/Education_LP0014.asp
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Remarks
Please write suggestions and improvements to be made in the project.
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