Final answer:
The girl's maximum speed on the swing can be calculated using the conservation of mechanical energy principle, resulting in approximately 7.67 m/s.
Step-by-step explanation:
The maximum speed of the girl on the swing can be found by using the conservation of mechanical energy principle. At the highest point, she has maximum potential energy and zero kinetic energy. At the lowest point, she has minimum potential energy and maximum kinetic energy. Since energy is conserved, the potential energy difference between the highest and the lowest points is converted into kinetic energy at the lowest point.
The potential energy (PE) at the highest point can be calculated by using PE = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s2), and h is the height difference (3.8 m - 0.8 m = 3.0 m). The kinetic energy (KE) at the lowest point is given by KE = 1/2 mv2, where v is the velocity which we are trying to find.
Setting the potential energy equal to the kinetic energy gives mgh = 1/2 mv2. Solving for v, the mass m cancels out and the equation simplifies to v = √(2gh). Plugging in the values, v = √(2 * 9.8 m/s2 * 3.0 m), we find that the maximum speed is approximately 7.67 m/s at the lowest point.