Affirmative and Negative Agreements

There are two types of agreements.
a. Affirmative Agreement
b. Negative Agreement
Observe the following structures:
a. Affirmative Agreements:
subject + auxiliary verb + too
so + auxiliary verb + subject
examples:
1. He is happy. I am happy.
He is happy and I am too.
He is happy and so am I.
2. She is working. They are working.
She is working and they are too.
She is working and so are they.
3. I need help. He needs help.
I need help and he does too.
I need help and so does he.
b. Negative Agreements:
neither + auxiliary verb + subject
subject + auxiliary verb + either
example:
1. She doesn’t smoke. He doesn’t smoke.
She doesn’t smoke and neither does he.
She doesn’t smoke and he doesn’t either.
2. I never watch TV. They never watch TV.
I never watch TV and neither do they.
I never watch TV and they don’t either.

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Embedded questions

Definition: Embedded questions are questions which are hidden between two sentences.
Note: Don’t put an auxiliary verb between the embedded question and the subject that comes after it.
Examples:
1. How old are you?
I am not sure how old I am.
2. When did the plane arrive?
Do you know when the plane arrived?
3. How long was the class?
I have no idea how long the class was.
4. Where does he live?
I don’t know where he lives.

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Intonation

Intonation is defined as a time or melody. It may also be called rise or fall in the speech sound or pitch change.
Types if intonation:
There are mainly 3 types of intonation. They are:
1. Rising intonation
2. Falling intonation
3. Mixed intonation

1.Rising intonation:
If our utterance (speech) ends with a high pitch, it marks intonation. The rising intonation can be used in the following Situations.
i. Yes/No – questions:
Is that your house?
ii. Repeated wh – question:
What? Who?
iii. Statement used as question:
Rita got married?
iv. Polite request:
Please, come in.
v. Greeting:
Good morning.
vi. Expression of Cordial gratefulness:
Thank you.

2. Falling intonation:
The falling intonation is usually marked by the fading voice at the end of the speech/utterance. We use it in the following cases
i. A simple statement:
Ravi wrote an essay.
ii. A simple compound:
Sit down.
iii. A simple wh - question:
What’s your name?
A simple expression of formality:
Thank you.

3. Mixed intonation:
- Rising + Falling
It is used in:
i. Yes/No – question containing two options connected by ‘or’
Will you have tea or coffee?
ii. Statement containing a list.
Hari saw a tiger, a bear, a monkey and a donkey.
iii. Negative command + question tag:
Don’t touch me, wil you?
iv. Sentence beginning with a dependent clause:
If you are ill, you can have a rest.
v. Sentence Containing the name of a person in the name of a person in direct address:
Mr. Thapa, this is my friend.

- Falling + Rising
It is used in:
i. Statement + question tag
It’s hot today, isn’t it?
ii. Affermative command + question tag
Do it, will you?

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Electrical Engineering Aspect

 Series and Parallel Circuit:
1. Series Circuit:
A simple circuit or path of any complex circuit in which same current passes through all the elements in that path is called series circuit. The general circuit diagram of series circuit using resistors is as shown as follows. Series circuit have following features.
- Same current passes through all the element.
i.e. I₁ = I₂= …………………… = I_N = I
- Voltage divides in all the components. So, It is known as voltage divider circuit.
i.e. V = V₁+ V₂+ …………… + V_N

i.e. Voltage across ith component is,V_i = (R_i/n)× V = {R_i/R_eq}× V
- Equivalent resistance of series circuit is given by sum of all the components.R_eq = R₁+ R₂+ …………….. + R_N

Note : For multiple number of inductors in series:
L_eq = L₁+ L₂+ ……………………….. + L_N 

For multiple number of capacitors in series
- If one components failsin performing it’s operation in series circuit, it cause failure of overall circuit due to opening of single path.
2. Parallel circuit :
When potential different across any two paths or components is same, those two paths or components are said to be parallely connected. The general circuit diagram of parallel circuit is as shown as follows.
Parallel circuit have following features :
- Current divides through different element so it is known as current divider circuit. Current in ith component is given by,
- Applied voltage across each elements remains same.
i.e. V₁= V₂ = V₃ = ………………. = V_N
- Equivalent resistance of parallel circuit is given by the formula.
Note : For multiple number of inductors in parallel
For multiple number of capacitors in parallel
Ceq = C₁+ C₂+……………… + C_N
- If one component fails in performing its operation in parallel circuit, it does not cause failure of any other circuit operation.
 

2. Energy Resources
Energy Resources are classified as follows.
- Renewable & Non-Renewable Resources
Renewal Resources :
They are not depleted and available in long term in periodic basis.
a) Biomass : Different types of plant materials which contains stored energy that comes from sun are called biomass. e.g. Gobar gas, wood, coal etc.
b) Solar energy : It is the form of energy which applies energy from the sun in form of solar radiation. e.g. Solar power, photovoltaic output energy.
c) Wind Energy : It is conversion of wind energy into a useful form of energy. e.g. Wind turbine and generator Output in electrical power, wind mills output in mechanical power.
d) Hydropower : power generated from energy in water is hydropower. e.g. Hydroelectricity.
e) Geothermal Energy : Energy from thermal energy generated and stored in the core of earth.
f) Ocean & Tidal Energy : They are generally those forms which generate electricity from the tides in the ocean.
g) Biofuel : Energy derived from biological carbon fixation. e.g. Biomethanol, Biodiesel etc.
Non Renewal Resources :     
They are depleted with continuous use.
a) Fossil Fuel : Fuels formed by natural processes such as anaerobic decomposition of buried dead organisms. E.g. Coal, Petrolium and natural gas.
b) Radioactive fuel/ Nuclear commercial Resources : Energy generated from nuclear reaction of radioactive material.
 

ii. Commercial and Non commercial Resources
Commercial Resources : They are useful for commercial processing purposes. e.g. Coal, Petrolium, Natural gas, Nuclear Energy, energy derived through thermo chemical and biochemical process etc.
Non commercial Resources : They are abundant in nature and are mostly used in rural area e.g. cowdung, charcoal, firewood, agricultural waste.

iii. Conventional and Non conventional Resources
Conventional Resources :
The sources of energy which have been in use for a long time. e.g. coal, petroleum, natural gas and water power.
Non conventional Resources :
The resources which are yet in the process of development over the past few years : e.g. solar, wind, tidal, biomass, biogas, geothermal.

3. Transformers :
- It is static electric machine which transfers electrical power from one circiut to another without changing frequency.
- It works on the principle of mutual induction.
- Parts of transformers.
a) Primary winding : Where input is given
b) Secondary winding : Where output is taken
c) Core : On which primary and secondary windings are wounded.
d) Yoke : Horizontal part of transformer core
e) Limb : Vertical part of transformer core
- It doesn’t work with DC source.
- Soft iron is used for making transformer core because of its high permeability.
- Transformation ratio (K) = N₂/N₁ = V₂/V₁ = I₁/I₂ 
a) If K = 1, N₁= N₂ (isolation transformer)
b) If K > 1, N₁< N₂ (step-up transformer)
c) If K < 1, N₁> N₂ (step-down transformer)
- Rms value of emf, induced in secondary winding is, E₂= 4.44fϕm
N₂
- Ideal transformer is that which has
a) No winding resistance
b) Core with infinite permeability
c) No coreloss and 100% efficiency.
- Eddy current is current circulating within core.
- Eddy loss α (I_eddy)².
- Copper loss in transformer is loss in winding resistance.
- During transformer operation, frequency remains constant.
- Core of transformer is laminated by varnish to reduce eddy current current loss.
- Capacity of transformer is measured KVA.
- Transformers are classified into three types according to utilities.

a) Power transformer :
• May be step-up or step-down (i.e. K >1 or K< 1).
• Rating is above 200 KVA.
• They are used at stations and substations.
b) Distribution transformer :
• They are step down (K<1).
• Used for consumer purposes.
• Rating upto 200 KVA.
c) Instrumental transformer :
• They are used for measurement purpose.
• CT (current transformer) is used to measure very high current.
• PT (potential transformer) is used to measure very high voltage.
 

4.Electrical Energy Generation
- It is based on the fundamental principles of electricity generation discovered by Michael Faraday.
- If Bm is maximum flux density, v is velocity of rotating coil, L is length of coil and ϕ is angle between axis of uniform magnetic field and axis of uniform magnetic field and axis of coil, magnetitude of induced emf is
E = BmLV sinϕ

Electricity Generation:
i. Convential methods of power Generation:
(They make use of prime mover for dividing electrical machines) e.g. Thermal, Hydro, Nuclear, Solar power generation etc.
ii. Non convential Methods of power Generation:
(They donot involve prime mover for power generation) e.g. solar cells, fuel cells, thermo electric generation etc.
 

Instrument In Electrical Measurement:
- Ammeter is used to measure current current through any element.
- Ideal ammeter has zero internal resistance.

- Ammeter is always connected in series with the element through which current is to be measured.
- Voltmeter is used to measure potential difference between two points of circuits.
- Voltameter is always connected in parallel with the element across which p.d. is to be measured.
- Ideal voltmeter has infinite resistance.
- Instrument voltmeter has infinite resistance.
- Instrument used to measure active electrical power is wattmeter i.e. It measures active power only i.e. VIcosφ.
 It has current coil which measures current (I) and voltage coil which measures p.d.(v) across  any element.

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Basic Trigonometry

Angle Measurement :
1. 1 radian angle (1ͨ) = 90 degrees (90°)
2. 1° = 60 minutes (60')
3. 1' = 60 seconds (60")
4. 1 right angle = 100 grades =100ᵍ
5. 1ᵍ = 100'
6. 180° = 200ᵍ = π radians
1 radians = 57° (approx.)
Trigonometric ratios:
1. (i) tanθ = sinθ/cosθ, Cotθ = cosθ/sinθ
(ii) sinθ.cosecθ = 1
(iii) cosθ.secθ= 1
(iv) tanθ.cotθ= 1
2. (i) sin²θ + cos²θ = 1
(ii) sec²θ – tan²θ = 1
(iii) cosec²θ – cot²θ  = 1
3. |sinθ| ≤ 1, |cosθ| ≤ 1, |cosecθ| ≥ 1, |secθ| ≥ 1 for all θ ϵ R.
Quadrant Rule :
90° = All trigonometric ratios are +ve
180° =  Sin and Cosec are +ve and others are –ve
270° =  Tan and cot are +ve and others are –ve
360° =  Cos and Sec are +ve and others are -ve

Trogonometric ratio of some Standard Angles :


1) Sin(90-θ) = cosθ
Cos(90-θ)= sinθ
Tan(90-θ)= cotθ
2) Sin(90+θ) = cosθ
Cos(90+θ)= -sinθ
Tan(90+θ)= -cotθ
3) For  n ϵ I
sin(n.360°+θ)=sinθ
cosec(n.360°+θ)=cosecθ
cos(n.360°+θ)=cosθ
sec(n.360°+θ)=secθ
tan(n.360°+θ)=tanθ
cot(n.360°+θ)=cotθ

Formulae for compound angles :-
i. sin(A+B)= sinA.cosB+cosA.sinB.
ii. sin(A-B) = sinA.cosB-cosA.sinB.
iii. cos(A+B)= cosA.cosB-sinA.sinB.
iv. cos(A-B)= cosA.cosB+sinA.sinB.
v. tan(A+B)= (tanA + tanB)/(1 – tanAtanB).
vi. tan(A-B)= (tanA – tanB)/(1 + tanAtanB).
vii. cot(A+B)= (cotA.cotB – 1)/(cotB + cotA).
viii. cot(A-B) = (cotA.cotB + 1)/(cotB – cotA).
ix. sin(A+B).sin(A-B) = sin²A – sin²B.
x.  cos(A+B) = cos²A-cos²B.
Multiple  and  submultiple  angles :-
i. sin2A = 2sinA.cosA = 2tanA/(1 + tan²A).
ii. cos2A = cos²A-sin²A = 1-2sin²A = 2cos²A-1 = (1 – tan²A)/(1 + tan²A).
iii. tan2A = 2tanA/(1 – tan²A .
iv. sin3A = 3sinA – 4sin³A.
v. cos3A = 4cos³A – 3cosA.
vi. tan3A = (3tanA – tan3A)/(1 – 3tan²A).
vii. cot3A = (cot³A – 3cotA)/(3cot²A – 1).
Changing  product  into  sum  or  difference  and  vice –versa :
1. 2sinA.cosB = sin(A+B) + sin(A-B).
2. 2cosA.sinB = sin(A+B) – sin(A-B).
3. 2cosA.cosB = cos(A+B) + cos(A-B).
4. 2sinA.cosB = cos(A-B) – cos(A+B).
5. sinC +sinD = 2sin{(C+D)/2}.cos{(C–D)/2}.
6. sinC – sinD = 2cos{(C+D)/2}.sin{(C–D)/2}.
7. cosC + cosD = {(C+D)/2}.sin{(C–D)/2}.
8. cosC – cosD = 2sin{(C+D)/2}.sin{(D–C)/2}.
Even function :-
A function of the form f(-x) = f(x) is known as even function. Eg.
sec(-θ) =secθ     cos(-θ)= cosθ
f(x) =x2, f(x) = |x| etc.
Note :-
i. Derivative of the even function is odd function.
Even function : f(-x) = f(x)
Differentiation: f '(-x) = -f '(x)  (Odd function)
ii. If x is replaced by –x in an even function
there is no change in the equation of the 
curve. i.e. f(x) = x²
putting x=-x , f(-x) = (-x) ² = x²
iii. Portion of the curve lying on the either side of x-axis are equal. i.e. even function is symmetric about y-axis.
Odd function : A function of the form f(-x)=-f(x) is known as odd function. e.g.: sin(-A)=-sinA, tan(-A)=-tanA, f(x) = x5, f(x) = xcosX etc.
Note :
i. Derivative of the odd function is an even function. i.e.
f(-x) = -f(x)
Differentiating:
f'(-x)(-1)=-f'(x)
f'(-x)=f'(x)
i.e.f'(x) is an even function.
ii. Constant function is both even and odd function.
Periodic function : The function of the form f(x+p)=f(x), where p is least positive number is called periodic function.
For sinx, cosx, p =2π
For tanx, cotx, p = π
i.e. sin(2π + θ)= sinθ, cos(2π +θ) =cosθ, tan(π + θ) = tanθ
The periodic function of cos(ax+b) is 2π/a.
Tips:
- π is an irrational number. π ≈ 22/7 =3.1414…….
- In the result θ = l/r; θ is always in radians wheareas l and r have same units.
- Area of a sector = ½ r²θ
- The sum of interior angles of a polygon of n sides = (n-2) × 180˚
  sin²x + cosec²x ≥ 2.
  cos²x + sec²x ≥ 2.
  tan²x + cot²x ≥ 2.
  Above are true for every real x.
- Maximum and minimum values of T-ratios:
i. If y = asinx + bcosx +c, then c – √ (a²+ b²) ≤ |y| ≤ c + √ (a²+ b²), Maximum value of y =c + √(a²+ b²)  Minimum value of y = c - √(a²+ b²)
ii. Greatest and least values of (asinx ± bcosx) are √ (a²+ b²) and √ (a² - b²) respectively.
iii. Maximum value of sinx and cosx = 1, minimum value = -1
iv. Max. value of (sinx.cosx) = 1/2 and min. value =-1/2.

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Voice

Note: there are two voices in English (Active and Passive voice). In the active voice the door of the action is given important and in the passive voice the victim (object) is given prominence. The object which is acted upon gains significance and the doer recedes into the background.

Passive forms of active in English :

Simple Tense

Subject + (is/am/are) + past pacticles.

examples :

He often writes to me.

- I am often written to by him.

He teases them.

- They are tested by him.

Simple Past tense :

Subject + (was/were) + past particles.

examples :

Ram killed a snake.

- A snake was killed by Ram.

Sita ate the apples.

- The apples were eaten by Sita.

Future Tense :

Subject + (will/shall) + be + past particles.

examples :

They are repairing the bridge.

- The bridge is being repaired by them.

He is scolding us.

- We are being scolded by him.

Present Continuous Tense :

Subject + (is/am/are) + being + past participle.

examples :

They are repairing the bridge.

The bridge is being repaired by them.

He is scolding us.

We are being scolded by him.

Past Continuous Tense :

Subject + (was/were) + being + past particles

examples :

They were carrying the injured player off the field.

The injured player was being carried off the field.

Present perfect Tense :

Subject + (has/have) + bee + past participles.

I have returned all the books to the library.

- All the books have been returned to the library by me.

They have repaired the road.

- The road has been repaired by them.

Past perfect tense :

Subject + (had + been) + past participle.

examples :

She had taught me.

- I had been taught by her.

The farmer had branched the newly bought cattle.

- The newly bought bought cattle had been branched by the farmer.

Future Perfect Tense :

Subject + (will have/shall have) + been + past participle.

examples :

We will have completed our course within a few months.

- Our course will have been completed within a few months.

They will have read the novel by tommorow.

- The novel will have been read by them tommorow.

(Note : perfect continuous tenses are not used in passive).

Passive of different sentences :

1. Modals :

- When a sentence has (can, could, may, might, must, should, would, or ought to) then these modals themselves don’t change, ‘be’ is added before the main verb.

Examples :

I can read a book.

- A book can be read by me.

She ought to take medicine.

- The medicine ought to be taken by her.

- When ‘have’ is used after the modals in the active, ‘have been’ is used in the passive.

examples :

Ram should have told him.

- He should have told by Ram.

I may have eaten it.

- He should have been eaten by Ram.

- It may have been eaten by me.

- When infinitive is used in the active voice, ‘to be’ + past participle is used in the passive.

examples :

I am to write a letter.

- A letter is to be written by me.

He has to invite us.

- We have to be invited by him.

2. Interrogative :

While transforming active questions, two things must be remembered.

- The question form must be retained.

- The question mark should not be forgotten

examples :

Does she read novels ?

- Are novels read by her ?

Did Hari catch the bus ?

- Was the bus caught by Hari ?

Will she forgive us ?

- Shall we be firgiven by her ?

Can you carry the box ?

- Can the box be carried by you ?

How does the girl help the boy ?

- How is the boy helped by the boy ?

Who made the kite ?

- By whom was the kite made ?

- Who was the the kite made by ?

3. Infinitive construction after passive verbs :

(say, think, know, feel, find, understand, believe, considered, claim, agree, assume, acknowledge, estimate, presume, report, decide, hope, remember) take the following forms in the passive.

examples :

People say that he is a learned man.

- It is said that he is a learned man.

- He is said to be a learned man.

People know that he was a theif.

-It is known that he was a theif.

- He is known to have been a theif.

I saw him work.

- He was seen to work.

People think that she is working hard.

- She is thought to be working hard.

4. Imperative :

Draw the curtain.

- Let the curtain be drawn.

Do not make fun of the poor.

- Let the poor no be made fun of.

Please draw the figure.

- You are requested to draw the figure.

5. Gerund combination :

He recommended using bulletproof glass

- He recommended that bulletproof glass should be used.

6. Omission of “by”:

- When the action is more important than the doer.

examples :

People are destroying the jungle.

- The jungle is being destroyed.

They murdered the chief.

- The chief murdered the chief.

The radiologist will x-ray your hand.

- Your hand will be x-rayed.

We are building a new public library.

- A new public library is being built.

- When subject would be the indefinite pronoun “one” is used

One sees this sort of advertisement everywhere.

- This sort of advertisement is being seen everywhere

- When we don’t know or have forgotten who did the action we use someone or somebody

examples :

have Somebody has forgotten who d-id the action we complete

Someone has moved my car.

- My car has been moved by someone.

(Note : If the subject is people, they, we, somebody, someone, nobody, no one, everybody etc. we don’t normally use “by + agent”)

7. Sometimes a transitive verb takes two objects. Direct and Indirect object. :

When there is only object, it becomes the grammatical subject of the passive sentence, where as there are two objects either of them could become the grammatical subject in the passive form of the sentences.

examples :

My uncle offered me a gift.

- A gift was offered to me by my uncle.

- I was offered a gift by my uncle

(Note : The forms of ‘be’ cannot be changed to passive voice.)

examples :

He is a man. (cannot be changed to passive voice).

8. Preposition with passive verbs :

Some passive verbs take other prepositions instead of ‘by’ .

examples :

Know, Oblique, Marry, Please, Displace, Satisfy, Dissatisfy, Interest, Contain, Astonish, Surprise.

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Molecular weight And Mole

- “The molecular weight of substance is the relative mass of 1 molecule of it compared with 1/12 of the mass of an atom of carbon 12 isotope.” Molecular weight can be calculated by summing up the atomic weights of its constituent atoms. Eg. molecular weight of H₂SO₄ is 2 × 1 + 1 × 32 + 4 × 16 = 98

- According to Berzelius Hypothesis, “Equal volumes of all gases under the same conditions of temperature and pressure contain same number of atoms.”

- According to Avogadro’s Hypothesis, “Equal volume of all gases under the same conditions of temperature and pressure contain same no. of molecules.”

- Avogadro’s hypothesis leads us to following Important deductions.

1. Atomicity of elimentary gases.

2. Relationship between molecular weight and vapour density i.e. molecular weight of a gas is twice, it’s vapour density.

3. Gram molecular volume of gases i.e. 1 gram mole (molecular mass expressed in gram) of all gases occupies 22.4 litres at NTP. For example, 1 mole (32 gram) of oxygen or 1 mole (2 gram) of hydrogen at NTP occupies 22.4 litres.

4. 1 mole of any substance contains equal number of molecules called Avogadro’s number and is equal to 6.023 × 10²³.

5. Determination of molecular formula from volumetric composition.

6. Mole Concept : Molecular mass expressed in terms of gram is called gram molecular mass or in short mole. The mole of substance can be calculated as,

No of Mole =

Mass in gram

Molecular wt.

E.g. Calculate the no. of mole in 80 gm of oxygen.

Solution: No of Mole =

Mass in gram

Molecular wt.

= 80/32 = 2.5

Determination of molecular weight by victor Meyer’s method :

We know that,

Molecular weight = 2 × vapour density and

Vapour density =

weight V_cc of substance at NTP

Weight V_cc of hydrogen at NTP

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Projectile Motion

1. Projectile Motion : 

 For vertical displacement,

y = usinθ t - ½gt²……………… (i)

For horizontal displacement,

X = ucosθ t

t = x/ucosθ …………………. (ii)

From equation (i) and (ii)

y = usinθ.x/ucosθ – ½g(x/ucosθ).

y = x tanθ - g/(2u²cos²θ)

is similar to y = ax + bx² so the path of projectile is parabolic.

2. Time of flight (T) :

T = 2usinθ/g

Time of ascent and time of descent are same i.e. t = usinθ/g.

3. Maximum Height (H) :

H = u²sin²θ/2g.

4. Range of Projectile (R) :

R = u²sin2θ/g.

R_max = u²/g.

5. Maximum height for the projectile with maximum range :

When a projectile is fired at an angle of 45° then,

R_max = u²/g.

For maximum height

R_max = 4H

6. Two Angle of projection for same horizontal range :

A projectile is fired from ground with velocity u at an angle of θ = with horizontal then

Range (R₁) = u²sin2θ/g

For another angle of projection for same range with same velocity will be

Horizontal range (R₁) = u²sin2θ/g

= u²/g.sin(180°- 2θ)

= u²/g.sin{2(90°- θ)}

= u²/g.sin2θ= R₂

θ and (90°- θ) are the two angle of projection for a projectile with same range with same velocity.

7. Velocity and direction of projectile at any height :

The horizontal component of projectile remain constant through the motion but vertical component is accelerating. At any point P, at horizontal displacement y

Horizontal velocity (V_x) = ucosθ

Vertical velocity (V_y) is given by

V_y²= (usinθ)²- 2gy

or, V_y²= √(u²sin²θ- 2gy)

Resultant velocity (V) = √(V_x² + V_y²)

= √(u² – 2gy)

∴V = √(u² – 2gy)

For direction α be the direction of resultant with horizontal,

So,

tanθ = v_y/v_x = {√(u²sin²θ-2gy)}/{ucosθ}

KE of projectile of mass m is, KE = ½mv² = ½m(V_x²+ V_y²).

Some important tips :

- In projectile motion acceleration is due to gravity.

- The path of the projectile is called trajectory.

- Nature of trajectory is parabolic.

- Horizontal component of velocity is constant through out the motion.

- If a projectile is projected so that its range obtained is maximum.then maximum height attained by it is ^th of maximum range.

- If a person can throw maximum horizontal distance R₀, then he can throw maximum height R₀/2.

- Height is maximum if θ = 90°, H_max = R_max.

- At heighest point angle between velocity and acceleration is 90°.

- Velocity is minimum at highest point hence kinetic energy is minimum.

- If a projectile is thrown with speed u at angle θ with horizontal the projectile makes an angle ‘α’ with horizontal then its speed, v = ucosθ.Secα.

- Average velocity during time of ascent (i.e. average velocity when projectile is at highest point).

V_avg = u/2.√( 1 + 3cos²θ)

- Average velocity when projectile strikes to ground,

V_avg.= ucosθ

- Change in speed when projectile is at highest point,

Δ = u(1-cosθ)= 2usin²θ/2

- If two projectile are projected with same speed at different angles then for same speed at different angles then for same range,

i. Sum of angles of projection must be 90°

θ + α = 90°,

ii. H_θ/H_α = tan²θ or cot²α

iii. (t_f)_θ/(t_f)_α = tanθ = cotα

iv. R = ½g(t_f)_θ(t_f)_α ⇒ R = ½gt₁t₂:t₁t₂ ∝ R

v. R = 4√( H_θH_α) ⇒ R²∝ H₁H₂

vi. H₁+ H₂= u²/2g.

Horizontal projection from Height :

For vertical motion

y = ½gt²…………………….(i)

for horizontal motion

t = x/u ……………….(ii)

from equation (i) and (ii),

y = gx²/2u²

y = (g/2u²).x² is the equation of parabola.

For time of flight :

H = ½gT²

T = √(2h/g)

For horizontal range :

Range (R) = u.T = u√(2h/g)

Some important tips :

When a ball rolled off from top of staircase with horizontal velocity ‘u’ having width ‘b’ and height ‘h’ the ball hits n^th step then, n =

2hu²

gb²

- If a man hits the target, he should point his gun in a direction higher than the target.

- If a man fires his gun directly aimed towards monkey at height, at same instant monkey at height, at same instant monkey starts falling then bullet hits the monkey.

- If two bodies are projected horizontally from certain height with different velocities u₁ and u₂ in opposite direction then

i. Their velocities are perpendicular after time,

ii. Velocities of 1^st body and 2^nd body when their velocity are perpendicular,v₁ =√(u₁²+ u₁u₂) and v₂ = √(u₂²+ u₁u₂)

iii. Their position vectors are perpendicular of after time,

t =

2√(u₁u₂)

g

- If a ball is droped from height ‘h’ from the top of an inclined plane of inclination ‘α’, ball elastically collides the it again, strike the inclined plane at a distance.

S = 8hSinα.

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