A2-kg box is pushed to the right by a force of 4 n for a distance of 32 m. it has an initial velocity of 4 m/s to the right. note: since this problem gives the distance ∆x traveled, first look at kinetic energy and calculate the net work using wnet = f ∆x = ∆k. after the kinetic energy values are calculated, then calculate the momenta and impulse values. remember that on quiz and test problems, you will need to decide which values should be calculated first.

a) what is the initial momentum pi of the box?

8 kg m/s

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b) what is the impulse or change in momentum ∆p of the box?

m/s

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c) what is the final momentum pf of the box?

m/s

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d) what is the initial kinetic energy ki of the box?



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e) what is the change in kinetic energy ∆k of the box?



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f) what is the final kinetic energy kf of the box?



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g) how long ∆t does it take for the box to travel the distance of 32 m?



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a) 8 Kg m/s b) 16 Kg m/s c) 24 Kg m/s d) 16 J e) 128 J f) 144 J

              g) 4 s

Explanation:

a) As momentum by definition is the product of mass times the velocity (is a vector quantity), we can write in this case the following:

pi = m. v₀ = 2 Kg . 4 m/s = 8 Kg. m/s

b) In order to get the change in momentum, we need to get first the final speed of the object.

As we have the total distance travelled, and we could find the acceleration, we could use a kinematic equation to solve the question, but later we will need the kinetic energy, it would be better to apply the work-energy theorem, and calculate ΔK as the work done by external force F, as follows:

ΔK = F . d = 1/2 m (vf² - v₀²)

As we know F, d, m, and v₀, we can solve the equation above for vf:

vf = 12 m/s

So, we can compute the final momentum as follows:

pf = m. vf = 2 Kg. 12 m/s = 24 Kg. m/s

Finally, we can find the change in momentum, as the difference between the final momentum and the initial one, calculated in a):

Δp = pf - pi = 24 Kg. m/s - 8 Kg. m/s = 16 Kg. m/s

c) As we have already found, final momentum is as follows:

pf = m . vf = 2 Kg. 12 m/s = 24 Kg. m/s

d) By definition the initial kinetic energy of the box is as follows:

Ki = 1/2 m v₀² = 1/2. 2 Kg .4² m²/s² = 16 J

e) We can find the change in the kinetic energy taking directly the difference between the final and initial ones, as follows:

ΔK = Kf - Ki = 1/2. 2 Kg (12² - 4²) m²/s² = 128 J

f) From above, we have Kf = 1/2 m. vf² = 1/2 . 2 Kg. 12² m²/s² = 144 J

g) As we know the magnitude of F, and the value of m, we can find the acceleration (assumed constant) , applying Newton's Second Law, as follows:

Fext = m .a ⇒ a = F/m = 4 N / 2 Kg = 2 m/s²

Appying the definition of acceleration, we can solve for t, as follows:

t = (vf-v₀) / a = (12 m/s - 4 m/s) / 2 m/s² = 4 s