To find the maximum speed of the child on the swing, we can use the conservation of energy principle, which states that the total energy of a system is conserved.
At the highest point of the swing, the child has maximum potential energy and zero kinetic energy. At the lowest point of the swing, the child has maximum kinetic energy and zero potential energy. Therefore, the total energy of the system remains constant, and we can write:
PE = KE
where PE is the potential energy and KE is the kinetic energy.
The potential energy of the child on the swing can be calculated as:
PE = mgh
where m is the mass of the child, g is the acceleration due to gravity, and h is the height of the swing at the highest point. Since the swing hangs from 2.6-m-long chains, the height of the swing at the highest point is:
h = 2.6 m - 2.6 m cos(50°) = 1.32 m
Substituting the values, we get:
PE = (20 kg)(9.81 m/s^2)(1.32 m) = 258.2 J
At the lowest point of the swing, all the potential energy is converted to kinetic energy. The kinetic energy of the child on the swing can be calculated as:
KE = (1/2)mv^2
where v is the speed of the child at the lowest point. Substituting the values, we get:
KE = (1/2)(20 kg)v^2
Equating the potential and kinetic energies, we get:
PE = KE
mgh = (1/2)mv^2
2gh = v^2
v = sqrt(2gh)
Substituting the values, we get:
v = sqrt(2 × 9.81 m/s^2 × 1.32 m) = 4.06 m/s
Therefore, the maximum speed of the child on the swing is 4.06 m/s.