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Composition of motion

Problem : The position of a projectile projected from the ground is :

x = 3 t y = ( 4 t - 2 t 2 )

where “x” and “y” are in meters and “t” in seconds. The position of the projectile is (0,0) at the time of projection. Find the speed with which the projectile hits the ground.

Solution : When the projectile hits the ground, y = 0,

0 = ( 4 t - 2 t 2 ) 2 t 2 - 4 t = t ( 2 t - 2 ) = 0 t = 0 , t = 2 s

Here t = 0 corresponds to initial condition. Thus, projectile hits the ground in 2 s. Now velocities in two directions are obtained by differentiating given functions of the coordinates,

v x = đ x đ t = 3 v y = đ y đ t = 4 - 4 t

Now, the velocities for t = 2 s,

v x = đ x đ t = 3 m / s v y = đ y đ t = 4 - 4 x 2 = - 4 m / s

The resultant velocity of the projectile,

v = ( v x 2 + v y 2 ) = { 3 2 + ( - 4 ) 2 } = 5 m / s

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Problem : A projectile, thrown from the foot of a triangle, lands at the edge of its base on the other side of the triangle. The projectile just grazes the vertex as shown in the figure. Prove that :

Projectile motion

The projectile grazes the vertex of the triangle.

tan α + tan β = tan θ

where “θ” is the angle of projection as measured from the horizontal.

Solution : In order to expand trigonometric ratio on the left side, we drop a perpendicular from the vertex of the triangle “A” to the base line OB to meet at a point C. Let x,y be the coordinate of vertex “A”, then,

Projectile motion

The projectile grazes the vertex of the triangle.

tan α = A C O C = y x

and

tan β = A C B C = y R - x

Thus,

tan α + tan β = y x + y R - x = y R x R - x

Intuitively, we know the expression is similar to the expression involved in the equation of projectile motion that contains range of projectile,

y = x tan θ 1 - x R

tan θ = y R x R - x

Comparing equations,

tan α + tan β = tan θ

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Projectile motion with wind/drag force

Problem : A projectile is projected at angle “θ” from the horizontal at the speed “u”. If an acceleration of “g/2” is applied to the projectile due to wind in horizontal direction, then find the new time of flight, maximum height and horizontal range.

Solution : The acceleration due to wind affects only the motion in horizontal direction. It would, therefore, not affect attributes like time of flight or maximum height that results exclusively from the consideration of motion in vertical direction. The generic expressions of time of flight, maximum height and horizontal range of flight with acceleration are given as under :

T = 2 u y g

H = u y 2 2 g = g T 2 4

R = u x u y g

The expressions above revalidate the assumption made in the beginning. We can see that it is only the horizontal range that depends on the component of motion in horizontal direction. Now, considering accelerated motion in horizontal direction, we have :

x = R = u x T + 1 2 a x T 2

R = u x T + 1 2 g 2 T 2

R = R + H

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Problem : A projectile is projected at angle “θ” from the horizontal at the speed “u”. If an acceleration of g/2 is applied to the projectile in horizontal direction and a deceleration of g/2 in vertical direction, then find the new time of flight, maximum height and horizontal range.

Solution : The acceleration due to wind affects only the motion in horizontal direction. It would, therefore, not affect attributes resulting exclusively from the consideration in vertical direction. It is only the horizontal range that will be affected due to acceleration in horizontal direction. On the other hand, deceleration in vertical direction will affect all three attributes.

1: Time of flight

Let us work out the effect on each of the attribute. Considering motion in vertical direction, we have :

y = u y T + 1 2 a y T 2

For the complete flight, y = 0 and t = T. Also,

a y = - g + g 2 = - 3 g 2

Putting in the equation,

0 = u y T - 1 2 X 3 g 2 X T 2

Neglecting T = 0,

T = 4 u y 3 g = 4 u sin θ 3 g

2: Maximum height

For maximum height, v y = 0 ,

0 = u y 2 2 X 3 g 2 X H

H = u y 2 3 g = u 2 sin 2 θ 3 g

2: Horizontal range

Now, considering accelerated motion in horizontal direction, we have :

x = R = u x T + 1 2 a x T 2

R = u x 4 u y g + 1 2 g 2 4 u y g 2

R = 4 u y g [ u x + 1 2 g 2 4 u y g ]

R = 4 u y g { u x + u y }

R = 4 u 2 sin θ g [ cos θ + sin θ ]

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Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Kinematics fundamentals. OpenStax CNX. Sep 28, 2008 Download for free at http://cnx.org/content/col10348/1.29
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