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Solution of Assignment No. ( 3 ) - 1 Solution of Assignment...

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1 Solution of Assignment No. ( 3 ) : Kinematics of plane motion Cartesian coordinates 1. The parametric equations of the plane motion of a particle are : x = 16t , y = 12t – 5t 2 where x, y are in meters and t is in seconds . Determine : a–the time, the position and the velocity when the particle’s path intersects the horizontal x–axis b – the maximum height of the path above the horizontal x – axis and the velocity of the particle at the maximum height . SOLUTION x = 16 t (1) y = 12 t – 5 t 2 (2) Differentiating twice w.r.to time the equations (1) & (2) to get the velocity and acceleration components: 16 x = & (3) t 10 12 y - = & (4) 0 x = & & (5) 10 y - = & & (6) Acceleration analysis: const f . const 0 9 - 0 10 x y tan , const. m/s 10 y x f 2 2 2 = = ° = - = = = = = + + + = r & & & & & & & & α α Maximum height above x-axis : The particles attains maximum height when its vertical velocity components vanishes, i.e. at 0 = y & , so in equation (4) : 0 = 12 – 10 t t = 1.2 sec This time is the time of reach to the highest point, in equation (2): max y = 12 * 1.2 – 5*(1.2) 2 = 7.2 m
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2 In equations (3) & (4): 0 , 0 x y tan , s / m 16 x 0 y , s / m 16 x = = = = = = = α α υ & & & & & The positions of fall on x-axis: These positions are at y = o , so substituting y = o in equation (2) , we can determine the instant of time (time of flight) passed till the path intersects the x-axis: then in equation (1) we can determine the x-coordinate: o 1 2 2 37 ) 16 12 ( tan , s / m 20 12 16 v 12 - 10(2.4) - 12 y , m/s 16 x m 38.4 x 16(2.4) x sec 2.4 t , 0 t ) t 5 12 ( t 0 t 5 t 12 0 y , 2 - = - = = + + + = = = = = = = = - = - = = - ψ & & 38.4m x y y max =7.2m B A s / m 20 v B = o 37
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3 2. The parametric equations of a particle` s path are given by : x = a cos 2 t , y = a sin 2 t where a is constant .Determine : a – the initial conditions .
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