A dart of mass m is accelerated horizontally through a tube of length L situated a height h above the ground by a constant force F. Upon exiting the tube, the dart travels a horizontal distance Δx before striking the ground, as depicted in the diagram attached.

(a) Develop an expression for the velocity of the dart, v, as it leaves the tube in terms of Δx, h, and any fundamental constants.

(b) Derive an expression for the kinetic energy of the dart as it leaves the tube in terms of m, Δx, h, and any fundamental constants.

(c) Derive an expression for the work done on the dart in the tube in terms of F and L

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(d) Derive an expression for the height of the tube above the ground in terms of m, Δx, L, F, and any fundamental constants.