Question

A cone falling with a speed V0strikes and penetrates the block of a packaging material. The acceleration of cone after impact is a=g-cx^{2. Where c is a positive constant and x is the penetration distance . If maximum penetration depth is x metre then find c.}

Answer 1

Discription of every step given;

The acceleration of the second time derivative of the distance traveled:
a = d²x/dt² = g - c·x²

This is a second order ODE consisting only of the function x and its derivatives but not the variable (t) on which the function depends. For such an ODE you can reduce the order by substituting the first derivative:

v = dx/dt => d²x/dt² = d/dt(dy/dt) = d/dx(dx/dt) · dx/dt = dv/dx · v

The physical meaning of the first derivative is the speed of the cone.

Hence:
v·dv/dx= g - c·x²


separate variables and integrate
=>
∫ v dv = ∫ g - c·x² dy
=>
v²/2 = g·x - c·x³/3 + d
(d is the constant of integration)

This equation looks a bit unusual and it is tricky to solve for x(t). However it is sufficient to solve this problem.

You can evaluate d from the initial condition:
at x=0 the velocity of the cone is v=v₀
=>
v₀²/2 = g·0 - c·0³/3 + d
=>
d = v₀²/2
=>
v²/2 = g·x - c·x³/3 + v₀²/2


The second condition is: when the cone reaches its maximum depth Xm he comes to rest.
at x=Xm v=0
<=>
0²/2 = g·Xm - c·Xm³/3 + v₀²/2
=>
c = 3·(g·Xm - v₀²/2) / Xm³

Thank you :)