A 31.0 kg child on a swing reaches a maximum height of 1.92 m above their rest position.
Assuming no loss of energy:
a) At what point during the swing will she attain their maximum speed?
b) What will be her maximum speed through the subsequent swing?
c) Assuming this maximum height was the result of one push from her parent, what was the
PE = 31 * 9.81 * 1.92 Joules
at bottom (1/2) m v^2 = PE
suspect C wants work done which is the same as the PE again
To answer these questions, we can use the principle of conservation of mechanical energy. The total mechanical energy of the child on the swing remains constant throughout the swing since there is no loss of energy. The mechanical energy of the system is the sum of kinetic energy and potential energy.
a) To find the point during the swing when the child attains maximum speed, we need to determine the point where all the potential energy is converted into kinetic energy. This occurs when the child is at the lowest point of the swing. At the lowest point, all the potential energy is converted into kinetic energy, resulting in maximum speed.
b) To find the maximum speed, we can use the conservation of mechanical energy equation:
Potential Energy + Kinetic Energy = Constant
At the highest point, when the child reaches a maximum height, the potential energy is at its maximum, and the kinetic energy is zero. The potential energy at this point can be calculated using the formula:
Potential Energy = m * g * h
Where m is the mass of the child (31.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the maximum height reached (1.92 m).
So, Potential Energy = (31.0 kg) * (9.8 m/s^2) * (1.92 m) = 576.7 Joules
Since the total mechanical energy is constant, the kinetic energy at the lowest point (maximum speed) will be equal to the potential energy at the highest point.
Kinetic Energy = 576.7 Joules
The formula for kinetic energy is:
Kinetic Energy = (1/2) * m * v^2
Where m is the mass of the child (31.0 kg) and v is the velocity (speed).
Rearranging the equation, we can solve for v:
v^2 = (2 * Kinetic Energy) / m
v^2 = (2 * 576.7 Joules) / 31.0 kg
v^2 = 37.168 Joules / kg
Taking the square root of both sides, we find:
v = √(37.168 Joules / kg) = 6.091 m/s
Therefore, the child's maximum speed through the subsequent swing is approximately 6.091 m/s.
c) To find the work done by the parent in pushing the child, we can use the work-energy principle. The work done by the parent is equal to the change in the child's mechanical energy.
Initial Mechanical Energy = Potential Energy
Final Mechanical Energy = Kinetic Energy + Potential Energy
The work done by the parent is equal to the change in mechanical energy:
Work = Final Mechanical Energy - Initial Mechanical Energy
Since the child starts at rest, the initial mechanical energy is zero.
Work = (Kinetic Energy + Potential Energy) - 0
Work = Kinetic Energy + Potential Energy
Based on the previous calculations, the work done by the parent can be determined as:
Work = 576.7 Joules + 576.7 Joules = 1153.4 Joules
Therefore, the work done by the parent in pushing the child to reach the maximum height of 1.92 m is approximately 1153.4 Joules.