Dynamics | Part IA, 2002

The position xx of the leading edge of an avalanche moving down a mountain side making a positive angle α\alpha to the horizontal satisfies the equation

ddt(xdxdt)=gxsinα\frac{d}{d t}\left(x \frac{d x}{d t}\right)=g x \sin \alpha

where gg is the acceleration due to gravity.

By multiplying the equation by xdxdtx \frac{d x}{d t}, obtain the first integral

x2x˙2=2g3x3sinα+cx^{2} \dot{x}^{2}=\frac{2 g}{3} x^{3} \sin \alpha+c

where cc is an arbitrary constant of integration and the dot denotes differentiation with respect to time.

Sketch the positive quadrant of the (x,x˙)(x, \dot{x}) phase plane. Show that all solutions approach the trajectory

x˙=(2gsinα3)12x12\dot{x}=\left(\frac{2 g \sin \alpha}{3}\right)^{\frac{1}{2}} x^{\frac{1}{2}}

Hence show that, independent of initial conditions, the avalanche ultimately has acceleration 13gsinα\frac{1}{3} g \sin \alpha.

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