1) Lagrangian of the mechanical system is Derive all the Euler-Lagrange equation

Question

1) Lagrangian of the mechanical system is Derive all the Euler-Lagrange equations (ELE) for this system. 2) Euler-Lagrange equation (ELE) is and the initial conditions are: q(0) 5 m; and goo) 3 m/s. Intergrate the given equation and derive two kinematic equations q(t) and 3) A child is standing at the distance L 10 m away from the school building. The child throws a baseball with the initial velocity that has the magnitude v 20 m/s and the angle a 450 with the horizontal direction. Few moments later, the baseball flies through the open window. what was the height H above the gorund where the window is located?

Explanation / Answer

2)

d^2(q)/dt^2 = 2

So, integrating: dq/dt = 2*t + k

At t = 0, dq/dt = -3 = 2*0 +k

So, k = -3

So, dq/dt = 2t - 3 <-----------answer

Again integrating : q = 2t^2/2 - 3t + c

At t = 0, q = 5m

So, q = 0^2 -3*0 + c = 5

So, c = 5

So, q = t^2 - 3t + 5 <------answer

3)

s = ut + 0.5at^2

For the motion in horizontal direction :

10 = 20*cos(45 deg)*t

So, t = 0.707 s

In vertical direction :

H = 20*sin(45 deg)*t - 0.5*9.8*t^2

= 20*sin(45 deg)*0.707 - 0.5*9.8*0.707^2

= 7.55 m

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