A12-pack of omni-cola (mass 4.30 kg) is initially at rest ona horizontal floor. it is then pushed in a straight line for1.20 m by a trained dog who exerts ahorizontal force with magnitude 39.0n.
(a) use the work-energy theorem to find the final speedof the 12-pack if there is no friction between the 12-pack and thefloor.
m/s
(b) use the work-energy theorem to find the final speed of the12-pack if the coefficient of kinetic friction between the 12-packand the floor is 0.30.
m/s

a)  W = 46.8 J  and b)   v = 3.84 m/s

Explanation:

The energy work theorem states that the work done on the system is equal to the variation of the kinetic energy

    W = ΔK = -K₀

a) work is the scalar product of force by distance

    W = F . d

Bold indicates vectors. In this case the dog applies a force in the direction of the displacement, so the angle between the force and the displacement is zero, therefore, the scalar product is reduced to the ordinary product.

    W = F d cos θ

    W = 39.0 1.20 cos 0

    W = 46.8 J

b) zero initial kinetic language because the package is stopped

    W - = -K₀

    W - fr d= ½ m v² - 0

    W - μ N d = ½ m v

   on the horizontal surface using Newton's second law

     N-W = 0

     N = W = mg

     W - μ mg d = ½ m v

    v² = (W -μ mg d) 2/m  

    v = √(W -μ mg d) 2/m

    v = √[(46.8 -  0.30 4.30 9.8 1.20) 2/4.3 ]

    v = √(31.63 2/4.3)

    v = 3.84 m/s

3. dark matter

empty space could also be a simple vacuum