(a) The potential energy for the child just as she is released is equal to the potential energy at the bottom of the swing. Why? Because the potential energy is given by the formula mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above some reference point. At the top of the swing, the height is 2.20 m, and at the bottom of the swing, the height is 0 m. Therefore, the potential energy at the top of the swing is mgh = (25 kg)(9.81 m/s^2)(2.20 m) = 544 J, and the potential energy at the bottom of the swing is mgh = (25 kg)(9.81 m/s^2)(0 m) = 0 J. So the potential energy for the child just as she is released is 544 J, which is the same as the potential energy at the bottom of the swing. (b) To find the speed of the child at the bottom of the swing, we can use the conservation of energy. The total energy of the system (child + Earth) is conserved, so the potential energy at the top of the swing is converted into kinetic energy at the bottom of the swing. The formula for kinetic energy is KE = (1/2)mv^2, where m is the mass of the object and v is its speed. So we can set the potential energy at the top of the swing equal to the kinetic energy at the bottom of the swing: mgh = (1/2)mv^2 Solving for v, we get: v = sqrt(2gh) where sqrt means square root. Plugging in the numbers, we get: v = sqrt(2(9.81 m/s^2)(2.20 m)) = 6.3 m/s So the child will be moving at a speed of 6.3 m/s at the bottom of the swing. (c) To find the work done by the tension in the ropes, we need to use the work-energy principle. The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the tension in the ropes is equal to the change in the kinetic energy of the child as she swings from the initial position to the bottom of the swing. We can use