We can estimate the distance that material ejected from a growing crater cavity

Question

We can estimate the distance that material ejected from a growing crater cavity will travel using the ballistic equation. Remember from your introductory mechanics course that the equations of motion for a particle moving in the x-y plane are given by

??=??0+????0??+12??????2

??=??0+????0??+12??????2

where the subscript 0 refers to the initial conditions of the particle, x and y are the final x-y positions of the particle, vx and vy are the components of the velocity in the respective x and y directions, ax and ay are the accelerations in the x and y directions, and t is the time it takes the particle to travel the distance. Assume that the initial position of the particle is at the origin and that the particle feels only the gravitational acceleration from the planet. Assume the particle is ejected at an angle ? measured from the (horizontal) surface with velocity v.

a) Draw of cross-sectional sketch of the crater and the path taken by the ejected particle. Label x0, y0 x, y, v, g, and ? on your sketch.

b) Derive relationships for vx and vy as a function of v and ?.

c) Derive the distance that the particle travels (i.e., the ballistic equation): ??=??2(sin2??)??

You can assume a flat surface (i.e., neglect any curvature of the planet). HINT: recall that 2sin??cos??=sin(2??)

Explanation / Answer

please specify equations of motion

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