A 32-kg girl is bouncing on a trampoline. During a certain interval after she leaves the surface of the trampoline, her kinetic energy decreases to 205 J from 435 J. How high does she rise during this interval? Neglect air resistance.

To calculate the height the girl rises during this interval, we need to use the principle of conservation of energy, which states that the total mechanical energy (kinetic energy + potential energy) of a system remains constant unless acted upon by external forces.

In this case, we know that the girl's kinetic energy decreases from 435 J to 205 J. The difference between the initial and final kinetic energy is equal to the change in potential energy.

The formula for potential energy is given by:

Potential Energy = mgh

Where:
m = mass (in kg)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height (in meters)

First, let's calculate the change in potential energy:

Change in Potential Energy = Final Potential Energy - Initial Potential Energy

Since the girl's kinetic energy decreases, the change in potential energy will be positive.

Change in Potential Energy = 435 J - 205 J = 230 J

Now, we can calculate the height using the formula for potential energy:

230 J = (32 kg) * (9.8 m/s^2) * h

Simplifying the equation:

230 J = 313.6 kg m^2/s^2 * h

Dividing both sides by 313.6 kg m^2/s^2:

h = 230 J / 313.6 kg m^2/s^2

h ≈ 0.733 m

Therefore, the girl rises approximately 0.733 meters during this interval.