To practice Problem-Solving Strategy 7.2 Problems Using Mechanical Energy II. The Great Sandini is a 60.0-kg circus performer who is shot from a cannon (actually a spring gun). You don’t find many men of his caliber, so you help him design a new gun. This new gun has a very large spring with a very small mass and a force constant of 1100 N/m that he will compress with a force of 4400 N. The inside of the gun barrel is coated with Teflon, so the average friction force will be only 40.0 N during the 4.00 m he moves in the barrel. At what speed will he emerge from the end of the barrel, 2.50 m above his initial rest position?

In the problems that have friction force this must be included in the energy conservation equation, in these cases the friction force performs dissipative work that is equal to the decrease in mechanical energy

[tex]X_{fr}[/tex] = [tex]E_{mf}[/tex] - [tex]E_{mo}[/tex]

Let's look for the initial mechanical energy

[tex]E_{mo}[/tex] = Ke = ½ k x²

Final energy

[tex]E_{mf}[/tex] = K + U = ½ m v² + mg y

The work of the rubbing force is

[tex]X_{fr}[/tex] = - fr d.

Let's look for the missing terms, let's start with the amount the spring compresses, let's use Hooke's law

F = - k x

x = F / k

x = 4400/1100

x = 4 m

Let's write the equation and calculate

-Fr d = (1/2 m v² + ½ mg y) - ½ k x²

½ m v² = -fr .d + 1 / 2k x² - ½ m g y

v² = 2/m (- fr .d + 1/2 k x² - ½ m g y)

v² = 2/60 (-40 4 + ½ 1100 4² - ½ 60 9.8 2.5)

v = √(7905/ 30)

v = 16.23 m/s²