Final answer:
To determine how high and how long it will take for the swing to reach maximum height after the 30 kg child jumps onto it, we use the conservation of momentum for the initial velocity calculation, and conservation of energy to find the maximum height. We then use the pendulum period formula to find the time to reach this height, which is one-quarter the period of the swing.
Step-by-step explanation:
The scenario described is an example of conservation of momentum followed by a conversion of kinetic energy into potential energy in a pendulum-like system. When the 30 kg child with an initial velocity of 2.5 m/s jumps onto the stationary swing where a 70 kg adult is sitting, the total momentum before the jump is equal to the momentum after the jump, due to the law of conservation of momentum.
First, let's find the initial momentum of the child: momentum = mass × velocity = 30 kg × 2.5 m/s = 75 kg·m/s. Since the swing is initially at rest, its initial momentum is zero. After the child jumps on the swing, the combined mass of the child and adult is 100 kg (70 kg + 30 kg), and they move together with the same velocity. To find this velocity, we use the conservation of momentum: 75 kg·m/s = 100 kg × v, resulting in an initial velocity v of the swing of 0.75 m/s.
With this initial velocity, we can use the conservation of energy to find how high they will swing. At the highest point of the swing, all the kinetic energy will have been converted into potential energy. Kinetic energy at the start is ½ × 100 kg × (0.75 m/s)² and the potential energy at the maximum height (h) is given by m × g × h (where g is the acceleration due to gravity, 9.8 m/s²). Equating the kinetic and potential energies, and solving for h, gives the maximum height they will reach.
Finally, the time to reach the maximum height can be found by considering the swing as a simple pendulum and using the period of a pendulum formula: T = 2π√(L/g), where L is the length of the ropes and g is the acceleration due to gravity. Since we are interested in the time to reach the maximum height, this will be a quarter of the period.