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Projectile Motion (College Level)

 



Projectile is a body thrown with an initial velocity in the vertical plane and then it moves in two dimensions under the action of gravity alone without being propelled by any engine or fuel.Its motion is called projectile motion.The path of a projectile is called its trajectory.
Examples:
  1. A packet released from an airplane in flight.
  2. A golf ball in flight.
  3. A bullet fired from a rifle.
  4. A jet of water from a hole near the bottom of a water tank.
Projectile motion is a case of two-dimensional motion .Any case of two dimensional motion can be resolved into two cases of one dimensional motion -one along the x-axis and the other along the y-axis.The two cases can be studied separately as two cases of one dimensional motion.The results from two cases can be combined using vector algebra to see the net result
What is important to remember is that the motion along the horinzontal direction does not affect the motion along the vertical direcion and vice versa.Horizontal motion and vertical motion are totally independent of each other .
A body can be projected in two ways :
  1. Horizontal projection-When the body is given an initial velocity in the horizontal direction only.
  2. Angular projection-When the body is thrown with an initial velocity at an angle to the horizontal direction.
We will study the two cases separately.We will neglect the effect of air resistance.We will take x-axis along the horizontal direction and y-axis along the vertical direction.
 

A body is thrown with an initial velocity u along the horizontal direction.We will study the motion along x and y axis separately.We will take the starting point to be at the origin.
Along x-axis
Along y-axis
1. Component of initial velocity along x-axis.
ux=u
1. Component of initial velocity along y-axis.
uy=0
2. Acceleration along x-axis
ax=0(Because no force is acting along the horizontal direction)
2.Acceleration along y-axisay=g=9.8m/s2It is directed downwards.
3. Component of velocity along the x-axis at any instant t.
vx=ux + axt
=u + 0
vx=u
This means that the horizontal component of velocity does not change throughout the projectile motion.
3. Component of velocity along the y-axis at any instant t.
v
y=uy + ayt
=0 + gt
vy=gt
4. The displacement along x-axis at any instant t
x=u
xt + (1/2) axt2
x=uxt + 0
x=u
t
4. The displacement along y-axis at any instant t
y= u
yt + (1/2) ayt2
y= 0 + (1/2) ayt2
y=1/2gt
2
Equation of a trajectory(path of a projectile)
We know at any instant x = ut
t=x/u
Also, y= (1/2)gt2
Subsituting for t we get
y= (1/2)g(x/u)2
y= (1/2)(g/u2)x2
y= kx2 where k= g/(2u2 )
This is the equation of a parabola which is symmetric about the y-axis.Thus,the path of projectile,projected horizontally from a height above the ground is a parabola.


We know ,at any instant t
vx= u
v
y= gt
v= (v
x2+ vy2)1/2 = [u2 + (gt)2]1/2
Direction of v with the horizontal at any instant :
(angle) = tan-1 (vy/vx)= tan-1 (gt/u)
 
Time of flight (T):It is the total time for which the projectile is in flight ( from O to B in the diagram above)
To find T we will find the time for vertical fall
From
y= uyt + (1/2) gt2When , y= h , t=T
h= 0 + (1/2) gt2
T= (2h/g)1/2
 
Range (R) :It is the horizontal distance covered during the time of flight T.
From x= ut When t=T , x=R
R=uT
R=u(2h/g)1/2
 
We will now consider the case when the object is projected with an initial velocity u at an angle to the horizontal direction.
We assume that there is no air resistance .Also since the body first goes up and then comes down after reaching the highest point , we will use the Cartesian convention for signs of different physical quantities.The acceleration due to gravity 'g' will be negative as it acts downwards.
We will separate the motion into horizontal motion (motion along x-axis) and vertical motion (motion along y-axis) .We will study x-motion and y-motion separately.
X axis Y axis
1. Component of initial velocity along x-axis.
ux=u cosΦ
1. Component of initial velocity along y-axis.
uy=u sinΦ
2. Acceleration along x-axis
ax=0(Because no force is acting along the horizontal direction)
2.Acceleration along y-axisay= -g= -9.8m/s2(g is negative as it is acting in the downward direction)
3. Component of velocity along the x-axis at any instant t.
vx=ux + axt
=ucosΦ + 0= ucosΦ
vx=ucosΦ
This means that the horizontal component of velocity does not change throughout the projectile motion.
3. Component of velocity along the y-axis at any instant t.
v
y=uy + ayt
vy=usinΦ - gt
4. The displacement along x-axis at any instant t
x=u
xt + (1/2) axt2
x=ucosΦ.t
4. The displacement along y-axis at any instant t
y= u
yt + (1/2) ayt2
y= usinΦ.t - (1/2)gt2
Equation of Trajectory (Path of projectile)
At any instant t
x= ucosΦ.t
t= x/(ucosΦ)
Also , y= usinΦ.t - (1/2)gt2
Substituting for t
y= usinΦ.x/(ucosΦ) - (1/2)g[x/(ucosΦ)]
2
y= x.tanΦ - [(1/2)g.sec2.x2 ]/u2
This equation is of the form y= ax + bx2 where 'a' and 'b are constants.This is the equation of a parabola.Thus,the path of a projectile is a parabola .
Net velocity of the body at any instant of time t
vx=ucosΦ
v
y=usinΦ - gt
v= (v
x2+ vy2)1/2
Φ= tan-1(vy/vx) Where Φ is the angle that the resultant velocity(v) makes with the horizontal at any instant .
Angular Projectile motion is symmetrical about the highest point.The object will reach the highest point in time T/2 .At the highest point,the vertical component of velocity vy becomes equal to zero .
vy=usinΦ - gtAt t=T/2 , vy= 0
0= usinΦ - gT/2
T= (2usinΦ)/g
Equation for vertical distance (y component)y= uyt - (1/2)gt2
At t=T/2 , y=H
H= usinΦ.T/2 - (1/2)g(T/2)
2
substituting T
H= usinΦ.usinΦ/g - (1/2)g(usinΦ/g)
2= (u2sin2)/g - (u2sin2)/2g
H= (u2sin2)/2g
Range is the total horizontal distance covered during the time of flight.
From equation for horizontal motion, x=uxt
When
t=T , x=R
R= u
xT = ucos.2usinΦ/g
= u
22sinΦcosΦ/g = u2sin2Φ/g
using 2sinΦcosΦ= sin2Φ
R= (u2sin2Φ)/g

Introduction to Space Physics - M. Kivelson, C. Russell







Description: Designed as a text for upper-level undergraduate and first-year graduate students, this volume attempts to establish the curriculum for the modern student entering the fledgling field of space physics. The book provides a broad, yet selective, treatment of the subject, covering virtually all aspects of space plasmas in the solar system. There are sections on the sun and solar wind, the magnetized and unmagnetized planets, and the fundamental process of space plasmas including shocks, plasma waves, ULF waves, wave-particle interactions, and auroral processes. In addition to emphasizing analysis, the authors also place importance on underlying phenomenology with extensive attention to observations.

Physics for Scientists and Engineers 6th ed. (College Text) - Serway and Jewett










Description:

This best-selling, calculus-based text is recognized for its carefully crafted, logical presentation of the basic concepts and principles of physics. PHYSICS FOR SCIENTISTS AND ENGINEERS, Sixth Edition, maintains the Serway traditions of concise writing for the students, carefully thought-out problem sets and worked examples, and evolving educational pedagogy. This edition introduces a new co-author, Dr. John Jewett, at Cal Poly Pomona, known best for his teaching awards and his role in the recently published PRINCIPLES OF PHYSICS, Third Edition, also written with Ray Serway. Providing students with the tools they need to succeed in introductory physics, the Sixth Edition of this authoritative text features unparalleled media integration and a newly enhanced supplemental package for instructors and students!

What are the uses of capacitor?


Ans:One of the most ubiquitous passive components used is the capacitor, found in nearly every electronic device ever made. Capacitors have a number of essential applications in circuit design, providing flexible filter options, noise reduction, power storage and sensing capabilities for designers.
Filter Applications

Combined with resistors, capacitors are often used as the main element of frequency selective filters. The available filter designsand topologies are numerous and can be tailored for frequency and performance by selecting the proper component values and quality. Some of the types of filter designs include:
High Pass Filter (HPF)
Low Pass Filter (LPF)
Band Pass Filter (BPF)
Band Stop Filter (BSF)
Notch Filter
All Pass Filter
Equalization Filter

Decoupling/By-Pass Capacitors - Capacitors play a critical role in the stable operation of digital electronics by protecting sensitive microchips from noise on the power signal which can cause anomalous behaviors. Capacitors used in this application are called decoupling capacitors and should be placed as close as possible to each microchip to be most effective, as all circuit traces act as antennas and will pick up noise from the surrounding environment.Decoupling and by-pass capacitors are also used in any area of a circuit to reduce the overall impact of electrical noise.

Coupling or DC Blocking Capacitors - Since capacitors have the ability to pass AC signalswhile blocking DC, they can be used to separate the AC and DC components of a signal. The value of the capacitor does not need to be precise or accurate for coupling, but it should be a high value as the reactance of the capacitor drives the performance in coupling applications.

Snubber Capacitors - In circuits where a high inductance load is driven, such as a motor or transformer, large transient power spikes can occur as the energy stored in the inductive load is suddenly discharged which can damage components and contacts. Applying a capacitor can limit, or snub, the voltage spike across the circuit, making operation safer and the circuit more reliable. In lower power circuits, using a snubbing technique can be used to prevent spikes from creating undesirable radio frequency interference (RFI) which can cause anomalous behavior in circuits and cause difficulty in gaining product certification and approval.

Pulsed Power Capacitors

At their most basic, capacitors are effectively tiny batteries and offer unique energy storage capabilities beyond those of chemical reaction batteries. When lots of power is required in a short period of time, large capacitors and banks of capacitors are a superior option for many applications. Capacitor banks are used to store energy for applications such as pulsed lasers, radars, particle accelerators, and railguns. A common application of the pulsed power capacitor is in the flash on a disposable camera which is charged up then rapidly discharged through the flash, providing a large pulse of current.

Resonant or Tuned Circuit Applications

While resistors, capacitors and inductors can be used to make filters, certain combinations can also result in resonance amplifying the input signal. These circuits are used to amplify signals at the resonant frequency, create high voltage from low voltage inputs, as oscillators, and as tuned filters. In resonant circuits, care must be taken to select components that can survive the voltages that the components see across them or they will quickly fail.

Capacitive Sensing Application

Capacitive sensing has recently become a common feature in advanced consumer electronics devices, although capacitive sensors have been used for decades in a variety of applications for position, humidity, fluid level, manufacturing quality control and acceleration. Capacitive sensing works by detecting a change in the capacitance of the local environment through a change in the dielectric, a change in the distance between the plates of the capacitor, or a change in the area of a capacitor.

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