Mass fired up a ramp from a spring loaded gun?

Question

Can't quite figure this one out.

A block of mass m is placed in a smooth-bored spring gun at the bottom of the incline so that it compresses the spring by an amount x_c. The spring has spring constant k. The incline makes an angle theta with the horizontal and the coefficient of kinetic friction between the block and the incline is Mu. The block is released, exits the muzzle of the gun, and slides up an incline a total distance L .

Find , the distance traveled along the incline by the block after it exits the gun. Ignore friction when the block is inside the gun. Also, assume that the uncompressed spring is just at the top of the gun (i.e., the block moves a distance while inside of the gun). Use g for the magnitude of acceleration due to gravity.

Express the distance L in terms of x_c, k, m, g, Mu, and theta .


Approach I took:

K1 + U1 + U_grav = K2 + U2 + U_grav

0 + .5kd^2 + 0 = 0 + 0 + (mu)mgDsin(theta)

D = (kd^2) / 2((Mu)mgsin(theta)

Incorrect, however. It's exactly like another problem we did in class but that one did not have friction as a variable. So I'm sure that's where my mistake lies. Any ideas?

Answer 1

Looks like you just didn't properly account for frictional energy loss

The energy of the spring
.5*k*(x_c)^2
will be converted to potential energy gain up the incline
sin(th)*m*g*L
and the loss due to friction. To compute friction, first determine the normal force
N=cos(th)*m*g
the frictional force is
N*Mu
or
cos(th)*m*g*Mu

and the work done is the force times displacement
cos(th)*m*g*Mu*L
since the friction works parallel to the surface of the incline, it works over the entire distance L

Set up the equation
.5*k*(x_c)^2=
sin(th)*m*g*L+
cos(th)*m*g*Mu*L

do some algebra
.5*k*(x_c)^2=
=m*g*L*(sin(th)+cos(th)*Mu)
L=
(.5*k*(x_c)^2)/
(m*g*(sin(th)+cos(th)*Mu))

j