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\includegraphics[max width=\textwidth, alt={}, center]{f14ab5b6-f9eb-46a5-b5a6-3d34433313c6-10_326_938_262_609} A slide in a playground descends at a constant angle of \(30 ^ { \circ }\) for 2.5 m . It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg , modelled as a particle \(P\), starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section. The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.

1. It is given that the sloping section of the slide is smooth.
   1. Find the speed of the child when she reaches the bottom of the sloping section.
   2. Find the distance that the child travels along the horizontal section of the slide before she comes to rest.
2. It is given instead that the sloping section of the slide is rough and that the child comes to rest on the slide 1.05 m after she reaches the horizontal section. Find the coefficient of friction between the child and the sloping section of the slide.
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