Final answer:
To find the child's speed at the bottom of the swing, apply the conservation of mechanical energy, converting potential energy at the top to kinetic energy at the bottom. The height of the swing's arc is derived from the swing's length and angle, which allows for solving the speed using the equation for kinetic energy.
Step-by-step explanation:
To determine the child's speed at the bottom of the swing, we can use the principle of conservation of mechanical energy, assuming no energy is lost due to friction. When the child is at the top of the swing's arc, all of the swing's energy is potential energy (PE), which is converted into kinetic energy (KE) at the bottom of the swing.
At the top of the swing, the potential energy (PE) relative to the bottom point is given by:
PE = m * g * h
where:
m = mass of the child = 35.0 kg
g = acceleration due to gravity ≈ 9.81 m/s²
h = height of the swing's arc
The height (h) can be calculated using the length of swing l, which is 2.2 m, and the angle θ = 30° with the vertical:
h = l * (1 - cos(θ))
At the bottom of the swing, all this potential energy is converted to kinetic energy (KE):
KE = 0.5 * m * v²
where v is the speed.
Using the conservation of energy:
PE = KE
m * g * h = 0.5 * m * v²
The mass (m) cancels out from the equation, and we can solve for v:
v = √(2 * g * h)
After calculating h using the earlier equation and then substituting into this equation, we can find v, which is the speed of the child at the bottom of the swing.