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Mene ne elementary projectiles?

  • A karkashin ilmin kimiyyar lissafi wato physics, yau zamu yi karatunmu ne akan elementary projectile. Bayan mun fahimci me elementary projectile ya kunsa  zamu koyi lissafi akan time of flight, da maximum height and range tare da application of projectile. Sannan daga nan sai mu amsa tambayoyi wanda suke fitowa a jarabawar karshen gama sakandire musamman ma a WAEC, da NECO da kuma JAMB.

    A wannan makala ta mu zaku ga dayawan bayanai zanyi su ne a Turance sakamakon wadansu bayanan idan aka fassara baza’a gane ma’anarsu ba.

    Mene ne elementary projectile?

    A dimensional curved motion wanda ya ke dauke da constant acceleration, na particle wanda aka jefa shi a iska ana kiranshi da projectile motionyana da vertical downward acceleration due to gravity. Ko wane path na motion din yana zama parabola ne ko kuma a portion of a parabola.

    Misalan wannan motions din sune kamar haka:

    • A rebounced tennis ball thrown against a vertical wall
    • A stone shot from a catapult
    • The motion of a bullet fired from a gun.

    Wadannan sune misalan motions din, idan muka tsaya a saman gini muka wurgo kowanne irin object horizontally over the building, wannan object din da muka wurgo, it traces the same type of curved path as the rebounced tennis ball.

    Yanzu zamu dauko definition na projectile a Turance gashinan kamar haka:

    A projectile is any object thrown or released into space that travels along a curved or parabolic path. At any point in time a projectile motion is characterized by:

    1. Horizontal motion at constant velocity where linear acceleration, a = 0m/s2
    2. Vertical motion (up or down) with constant acceleration where gravitational acceleration, g = 9.8m/s2

    Lissafin projectile yana zuwa ne yawanci ta hanya guda biyu, ga su kamar haka:

    1. Objects Projected Horizontally from a Height.

    Fig 1.1

    Note: Abubuwan da yakamata musani dangane da object projected horizontally su ne, initial horizontal velocity, U, baya canzawa kwata kwata yana zama a constant din shi a lokacin motion. Initial vertical velocity, V = 0, however, the subsequent vertical velocities, V,V2…..Vn  yana canzawa with time.

    Ana amfani da equation na uniformly acceleration motion a irin wannan case din kamar yadda kuka gani a diagram din daya gabata. (v = u + at, v2 = u2 + 2as, s = ut + ½ at2 ), ina baku shawarar ku duba makala ta mai dauke da motion under gravity.

    2. Objects Projected at an Angle, , to the Horizontal.

    Fig 1. 2

    Note: Abubuwan da ya kamata mu sani dangane da objects projected at an angle, θ, to  the horizontal  sune, at a point na projection ( t = 0 ), the intial velocity U, can be resolved  in to the vertical component, Vy = U sin θ,  and the horizontal component, Vx = U cos θ. The horizontal component of the initial velocity, U cos θ remains the same  throughout the time of flight ( from t = 0 to t = 4s ). In contrast, vertical component na initial velocity ( Usinθ ) bayan projection yana canzawa ne tare da lokaci (changes with time) bayan time t, velocity yana koma wa Vy = Usinθ – gt, during upward motion Vy = Usinθ + gt, during down ward motion. Haka yana nuna cewa value na Vy ya dangantane da tsawon lokacin (t) da projectile din ya dauka a iska.

    Bayan projection a cikin iska zuwa sama, motion na projectile is governed by the resultant (V) of the vertical and horizontal components of the initial velocity as illustrated below. Fig 1.3

    • During upward motion (ii) during downward motion

     A kowanne yanayi, the resultant velocity,V is found using the pythogoras theorem.

    V2 =   +  

    Therefore, Resultant  velocity , V =  

    The resultant velocity forms an angle  with the horizontal plane and is calculated as follows

       tanθ =  Vy / Vx  therefore θ = tan-1  ( Vy / Vx )

    for a projectile in motion as shown in 1.2 the following applies.

    Time taken to reach maximum height, t = U sin θ / g

    Time of flight,T = 2U sin θ / g

    Maximum height, H = U2sin2θ /2g

    Horizontal Range,R = U2sin2θ /g

    Yanzu zamu dauko misalai mu amsa tambayoyin tare da wadannan fomuolin don kara koyon yadda ake lissafinsu.

    Misali na daya:

    An object is projected horizontally from the top of a cliff with a velocity of 7ms-1 and lands on the ground level at a point 56m from the base of the cliff. Calculate the height of cliff.

    Amsar tambaya:

    Abubuwan da aka bamu sune;

    Horizontal velocity V = 7ms-1

    Horizontal distance d = 56m

    Munsan cewa a velocity = distance / time, time taken by object to land on the ground is

    t = horizontal distance / horizontal velocity = 56/7 = 8s

    ana amfani da equation of motion  under gravity wajan neman height; h = ut + ½ gt2  

    substitute initial velocity u = 0ms-1 ( not 7ms-1 ); g =10ms-1 ; time t = 8s

    therefore , h -= 0 ×8 + ½ + 10 × 82 = 0.5 × 64 = 32m

    Misali na biyu:

    A ball is projected horizontally from a height of 20m above the ground  with an initial velocity of 0.4m/s. calculate the horizontal distance moved by the ball before hitting the ground. (g = 10ms2 )           WAEC 2005

    Amsar tambaya:

    Data:

    Given; height, h = 20m; initial horizontal velocity, u = 0.4m/s;

    Initial vertical velocity, u = 0

    The time taken to reach the ground is governed  by gravity, therefore, equation of motion under gravity is used  and also initial  velocity, u = 0

    Yanzu zamu dauko fomula da zamu yi amfani da ita a ciki sai musa abubuwan da aka bamu mu yi lissafinsu.

    Substituting into,h = ut + ½ gt2

     20 = 0 × t + ½ × 10 × t2

    20 = 5t2

    20 / 5 = t2

    t2 = 4 , therefore, t  = 2s

    Bayan mun yi lisssafin lokacin da ya dauka kamin ya yi reaching ground, the horizontal distance, s, can be calculated using equation of uniformly accelerated motion. Remembering that  acceleration, a = 0 and u = 0.4m/s

    substituting into s = ut + ½ at

    = 0.4 × 2 + ½  × 0 × 22

     = 0.8 + 0

    = 0.8m

    Yana da kyau dalibai  su kara kokarin kwatanta lissafin projectiles tare da amfani da fomula da muka bayar sannan su kara bincike game da wannan makala don samun karin ilmi a kai.

    Mai karatu na iya duba: Darasi game da electrical method

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