A 50- kg child sits in a swing suspended with 3.8- m-long ropes. The swing is held aside so that the ropes make an angle of 48 o with the vertical. Use conservation of energy to determine the speed the child will have at the bottom of the arc when she is let go.

To determine the speed the child will have at the bottom of the arc when she is let go, we can use the principle of conservation of energy.

First, let's define some variables:
m = mass of the child (50 kg)
L = length of the ropes (3.8 m)
θ = angle between the ropes and the vertical (48 degrees)

The potential energy of the child at the initial position can be calculated using the formula:
PE_initial = m * g * h

where g is the acceleration due to gravity (9.8 m/s^2) and h is the vertical displacement of the child from the initial position to the bottom of the arc. Since the child is held aside, the initial vertical displacement h can be found using trigonometry:
h = L * sin(θ)

Now, the total mechanical energy at the initial position is the sum of potential energy and kinetic energy:
E_initial = PE_initial + KE_initial

At the bottom of the arc, the potential energy is zero since the child is at ground level. Hence, the total mechanical energy at the bottom of the arc is equal to the kinetic energy:
E_bottom = KE_bottom

According to conservation of energy, the total mechanical energy remains constant:
E_initial = E_bottom

So, we can equate the two equations and solve for the speed at the bottom of the arc (v_bottom):
m * g * h = (1/2) * m * v_bottom^2

Canceling out the mass (m) and simplifying the equation, we get:
g * h = (1/2) * v_bottom^2

Now, substitute the values:
g = 9.8 m/s^2
h = L * sin(θ)
v_bottom = speed at the bottom of the arc (what we need to find)

Calculating:
v_bottom^2 = 2 * g * h
v_bottom^2 = 2 * 9.8 * (3.8 * sin(48))
v_bottom^2 ≈ 180.56

To find the speed (v_bottom), we take the square root of both sides:
v_bottom ≈ √180.56
v_bottom ≈ 13.43 m/s

Therefore, the child will have a speed of approximately 13.43 m/s at the bottom of the arc when she is let go.