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Two Dimension: Projectile Motion

BASIC CONCEPTS
1. Projectile Motion

Photo 10232   When a particle is so projected, that it makes certain angle with horizontal, then the motion of particle is said to be projectile. It is an example of curved motion. Examples : (a) A bullet is fired from a gun.

Photo 10984


 Assumptions
 Photo 10232 Path of projectile is a parabolic
 Photo 10232 acceleration due to gravity is constant.
 Photo 10232 Air resistance is negligible in a case of projectile motion hence time of ascend = Time of descent.

Calculation for projectile motion
 Photo 10232 If initial velocity of particle is u projected at angle  with horizontal. It has two components
    (i) u Photo 10997Photo 10230 horizontal component (Uniform)
    (ii) u Photo 10999Photo 10230 vertical component (Non-Uniform)
2. Time of ascent Photo 10230 It is time taken to reach maximum height (t)= Photo 10995


3. Time of flight Photo 10230 It is time taken in which body after describing a parabolic path again strikes the ground. Photo 11002   Photo 10232 time of ascent = time of descent = Photo 10995 
Photo 10232 time of flight is maximum when projectile is thrown vertically upwards.


4. Vertical component of projectile motion
ay = - g, Vyo = uPhoto 10999
At any instant of time t vertical component of velocity if
Vy = Vyo - gt = uPhoto 10999 - gt
vertical position of particle after time t
Photo 11004

5. Maximum height attained (H) Photo 10230 When particle reaches maximum height then Vy = 0
          Photo 11007
H is maximum when = 90, then maximum height attained is maximum Photo 11009

 

6. Horizontal range = H.V × time of flight

       Photo 11010
     when sin2 = 1, i.e. = 45º, R is maximum Photo 10232 Photo 11011  for = 45º

7. Velocity at top most point and velocity at the point of striking the ground
    (i) velocity at the topmost point Vx = uPhoto 10997, Vy = 0

    (ii) velocity at the point of striking the ground Vx = uPhoto 10997
    Vy = vertical component of velocity = –uPhoto 10999
    Magnitude of velocity = Photo 11017
    Photo 10232 projectile hits the ground with the same velocity making same angle with horizontal with which it was thrown.

8. Equation of trajectory of projectile
Photo 11018
Photo 10232 path of projectile is parabolic

9. Some important points concerning projectile motion.
    (a) the angle between velocity and acceleration during ascent of projectile if 180º < < 90º but during descent it is 90º < < 0º
    (b) incase of projectile motion at highest point
    Photo 10232 potential energy = Photo 11023 , K.E. = Photo 11024

    Photo 10232 When one body protected with same velocity in two different way but H.R. are same. In this case Photo 11025

    Photo 10232 If Photo 11027

    (c) during projectile motion weight of a body = 0

    (d) in the presence of air friction

    Photo 10232 the range and the maximum height attained are reduced

    Photo 10232 time of flight and angle of strike increases.

    At maximum height the radius of circular path
    Photo 11028

    If projectile thrown at angle then the time after which angle will lee Photo 11029
    Photo 11032
    Photo 11034
    Photo 11036Photo 11037


10. Motion on inclined plane
Photo 11049The motion of the projectile is a parabola. The velocity of the projectile along the inclined plane is : u cos Photo 11040 and the perpendicular to the plane is : u sin Photo 11040.The acceleration due to gravity perpendicular to the  inclined plane is : g cosPhoto 11041.

    (1) The time of flight of the projectile is given by:
    Photo 11042
    Remember: Photo 11043 and in this case.
    Photo 11044

    (2) The horizontal distance covered by the projectile will be given by:
    OC = (u cos Photo 11045) (T)
    Photo 11046

    (3) The distance covered by the projectile along the inclined plane is given by:
    Photo 11048
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