Use the work-energy theorem to find the speed of the sled at 7.0 m A child appli

Question

Use the work-energy theorem to find the speed of the sled at 7.0 m A child applies a force F parallel to the r-axis to a 10.0-kg sled moving on the frozen surface of a small pond. As the child controls the speed of the sled, the r -component of the force she applies varies with ther- coordinate of the sled as shown in figure (Figure 1) Suppose the sled is initially at rest at =0. You can ignore friction between the sled and the surface of the pond u= m/s Figure 1 of! Submit My Answers Give Up F (N) Incorrect; Try Again; 23 attempts remaining 10 Part B Use the work-energy theorem to find the speed of the sled at 11.0 m u- m/s x (m) 0 4 Submit My Answers Give Up

Explanation / Answer

The work done is equal to the area under the force-displacement curve:
work = F(x) dx

Here we have F(x) = 5x/4 for 0 m x 8 m
and F(x) = 10 - 5(x-8)/2 = 30 - 5x/2 for 8 m < x 12 m
where x is in meters and F in Newtons.

(i) at x = 7.0 m:
work = [0,7] 5x/4 dx = 5x²/8 |[0,7] = 5*7²/8 = 30.625 J
with no losses and no change in PE,
work = KE = 30.625 J = ½mv² = ½ * 10.0kg * v²
v = 2.475 m/s

(ii) at x = 11.0 m,
work = [0,8] 5x/4 dx + [8,11] (30 - 5x/2) dx
work = 5*8²/8 + (30x - 5x²/4) |[8,11]
work = 40 + 30(11 - 8) - (5/4)(11² - 8²) = 58.75 J
KE = 58.75 J = ½ * 10kg * v²
v = 3.43 m/s

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