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Motion in A Plane

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Physics Part I

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Physics

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Class 10 Class 12
A projectile is projected with velocity u at an angle θ with horizontal. Discuss the trajectory of projectile. Also find the velocity of projectile at any instant.

Let us consider a projectile is projected with velocity 'u' at angle 'θ' with horizontal direction from a point on the ground.

Let the point of projection be origin, right direction is positive direction of X-axis and vertically upward direction is positive direction of Y-axis.

Equation of trajectory:

Break the components of velocity and acceleration due to gravity in horizontal and vertical direction.

x0 = 0    ; yo = 0
ux = u cos straight theta   ; uy = u sin straight theta   
a= 0 ; ay = -g

Motion along horizontal direction:

straight x equals straight x subscript straight o plus straight u subscript straight x straight t plus 1 half straight a subscript straight x straight t squared equals straight u space cosθt            ...(1)

Motion along vertical direction:

space space space space straight y equals straight y subscript straight o plus straight u subscript straight y straight t plus 1 half straight a subscript straight y straight t squared

    equals usinθt minus 1 half gt squared                             ...(2)

Eliminating t from (1) and (2),

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This the required equation of trajectory.

Since the equation of trajectory is the equation of parabola, therefore projectile follows parabolic path.



Velocity of projectile:

Let at any instant,straight v with rightwards harpoon with barb upwards on top be the velocity of the projectile making an angle straight alpha with the horizontal. 

i.e., 

ux = u cos straight theta ,  uy = u sin straight theta 

ax = 0 , ay = -g 

Now, 

vx = ux + axt    ,     vy =  uy + ayt  

Therefore, 

vx = u cosstraight theta     , vy = u sinstraight theta - gt

The velocity f the projectile is given by, 

v = square root of straight v subscript straight x squared plus straight v subscript straight y squared end root
  equals space square root of straight u squared cos squared straight theta plus left parenthesis usinθ minus gt right parenthesis squared end root

equals space square root of u squared minus space 2 u g t space s i n theta plus g squared t squared end root
Angle pf projection is given by, tan straight alpha = fraction numerator straight u space sinθ space minus space gt over denominator straight u space cos space straight theta end fraction

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