Question

A block of mass m is placed in a smooth-bored spring gun at the bottom of an inclined plane, such that it compresses the spring by an amount x_c, as shown in the figure below. The spring has a spring constant k. The incline makes an angle θ with the horizontal and the coefficient of friction between the block and the inclined plane is μ. The block is released, exits the muzzle of the gun, and slides up the incline a total distance L before coming to rest. Express your answers in terms of known quantities. You can take the acceleration of gravity, g, to be known.

 a)  Model: How will you model this problem?

 b)   Visualize: Draw a before-and-after pictorial representation and list the known quantities in each state.

 c)  Solve: Taking the gravitational potential energy to be zero before the block is released, what is the total initial energy of the block?

 d)  Solve: What is the work done by friction on the block?

 e)  Solve: Find the energy the block has the instant after it travels a distance L up the ramp. Hint: do not forget to account for the distance the block travels while inside the spring gun.

 f)  Use energy conservation to express the distance L as a function of xc,k,m, g,μ, and θ.

 g) Assess: Verify that your units from part 

 f) have the proper dimensions of length.

Answer 1