From Fig. 7.37, initial speed is 6 m s-1. The trapezoid area gives work done as 150 J, raising kinetic energy to 330 J and final speed to 8.12 m s-1. Acceleration is never negative since force stays positive.
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Gravity does negative work upward and positive work downward. Initial kinetic energy is 400 J, while work done by gravity is minus 388 J. The work done by air resistance equals minus 12 J.
For rotational balance, effort times effort arm equals load times load arm. Since the adult has twice the weight of the child, the child must sit at twice the distance from the fulcrum compared to the adult.
Fuel consumed is proportional to kinetic energy gained. The first day total mass is 160 kg, and the second day total mass is 200 kg. The required ratio of fuel used is 160 to 200, which simplifies to 4 is ...
Arc length is (75/360) x 2 x (22/7) x 14 = (5/24) x 88 = 55/3 = 18.33 cm. Sector perimeter equals arc length plus two radii: 18.33 + 28 = 46.33 cm (or 46 and 1/3 cm).