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Problem : A bullet from a gun is fired at a muzzle speed of 50 m/s to hit a target 125 m away at the same horizontal level. At what angle from horizontal should the gun be aimed to hit the target (consider g = 10 m / s 2 ) ?

Solution : Here, horizontal range is given. We can find out the angle of projection from horizontal direction, using expression of horizontal range :

R = u 2 sin 2 θ g sin 2 θ = g R u 2 = 10 x 125 50 2 = 1 2 sin 2 θ = sin 30 0 θ = 15 0

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Problem : A projectile has same horizontal range for a given projection speed for the angles of projections θ 1 and θ 2 ( θ 2 > θ 1 ) with the horizontal. Find the ratio of the times of flight for the two projections.

Solution : The ratio of time of flight is :

T 1 T 2 = u sin θ 1 u sin θ 2 = sin θ 1 sin θ 2

For same horizontal range, we know that :

θ 2 = ( 90 0 - θ 1 )

Putting this, we have :

T 1 T 2 = sin θ 1 sin ( 90 0 - θ 1 ) = tan θ 1

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Problem : A projectile, thrown at an angle 15° with the horizontal, covers a horizontal distance of 1000 m. Find the maximum distance the projectile can cover with the same speed (consider g = 10 m / s 2 ).

Solution : The range of the projectile is given by :

R = u 2 sin 2 θ g

Here, R = 1000 m, g = 10 m / s 2 , θ = 15°

1000 = u 2 sin ( 2 x 15 0 ) g = u 2 2 g u 2 2 g = 1000 m

Now, the maximum range (for θ = 45°) is :

R max = u 2 g = 2000 m

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Maximum height

Problem : Two balls are projected from the same point in the direction inclined at 60° and 30° respectively with the horizontal. If they attain the same height, then the ratio of speeds of projection is :

Solution : Since the projectiles attain same height,

Projectile motion

H 1 = H 2 = H u 1 2 sin 2 60 0 2 g = u 2 2 sin 2 30 0 2 g u 1 2 X 3 4 = u 2 2 X 1 4 u 1 2 : u 2 2 = 1 : 3 u 1 : u 2 = 1 : 3

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Problem : A projectile is thrown vertically up, whereas another projectile is thrown at an angle θ with the vertical. Both of the projectiles stay in the air for the same time (neglect air resistance). Find the ratio of maximum heights attained by two projectiles.

Solution : Let u 1 and u 2 be the speeds of projectiles for vertical and non-vertical projections. The times of the flight for vertical projectile is given by :

T 1 = 2 u 1 g

We note here that the angle is given with respect to vertical - not with respect to horizontal as the usual case. As such, the expression of time of flight consists of cosine term :

T 2 = 2 u 2 cos θ g

As, time of flight is same,

2 u 1 g = 2 u 2 cos θ g u 1 = u 2 cos θ

On the other hand, the maximum heights attained in the two cases are :

H 1 = u 1 2 2 g H 2 = u 2 2 cos 2 θ 2 g

Using the relation u 1 = u 2 cos θ as obtained earlier, we have :

H 2 = u 1 2 2 g H 1 = H 2

The result is intuitive about the nature of projectile. The time of flight and vertical height both are consideration of motion in vertical direction. Since times of flight in both cases are same, the vertical components of two projectiles should be same. Otherwise, times of flight will be different. Now, if vertical component are same, then maximum heights have to be same.

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Height attained by a projectile

Problem : The times for attaining a particular vertical elevation during projectile motion are t 1 and t 2 . Find time of flight, T, in terms of t 1 and t 2 .

Solution : We can answer this question analytically without using formula. Let the positions of the projectile at two time instants be "A" and "B", as shown in the figure. The time periods t 1 and t 2 denotes time taken by the projectile to reach points "A" and "B" respectively. Clearly, time of flight, T, is equal to time taken to travel the curve OAB ( t 2 ) plus the time taken to travel the curve BC.

Projectile motion

Now, projectile takes as much time to travel the curve OA, as it takes to travel curve BC. This is so, because the time of travel of equal vertical displacement in either direction (up or down) in vertical motion under gravity is same. Since the time of travel for curve OA is t 1 , the time of travel for curve BC is also t 1 . Thus, the total time of flight is :

T = t 1 + t 2

Alternatively,

As the heights attained are equal,

h 1 = h 2 u y t 1 - 1 2 g t 1 2 = u y t 2 - 1 2 g t 2 2 u y ( t 2 - t 1 ) = 1 2 g ( t 2 2 - t 1 2 ) = 1 2 g ( t 2 + t 1 ) ( t 2 - t 1 ) t 2 + t 1 = 2 u y g = T

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Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
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the significance of food webs for disease transmission
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food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
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Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
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This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
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faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
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Innocent
cell is the structural and functional unit of life
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is the fundamental units of Life
Musa
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Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
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I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
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Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
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part of a tissue or an organ being wounded or bruised.
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Binomial nomenclature
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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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