A 27-kg girl is bouncing on a trampoline. During a certain interval after leaving the surface of the trampoline, her kinetic energy decreases to 200 J from 480 J. How high does she rise during this interval? Neglect air resistance.
ΔKE= ΔPE
480-200=mgh
h=(480-200)/27•9.8 = 1.06 m
To determine how high the girl rises during the interval, we need to make use of the principle of conservation of mechanical energy. The total mechanical energy of an object is the sum of its potential energy and kinetic energy.
In this case, when the girl is highest off the trampoline, all of her initial kinetic energy has been converted to potential energy. Thus, her initial kinetic energy is equal to her final potential energy:
Kinetic energy = Potential energy
480 J = mgh
Where:
m = mass of the girl (27 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height
Rearranging the equation to solve for h, we have:
h = 480 J / (m * g)
Substituting the given values:
h = 480 J / (27 kg * 9.8 m/s^2)
Evaluating the expression, we get:
h ≈ 1.80 meters
Therefore, the girl rises approximately 1.80 meters during the interval.
To determine the height the girl rises during this interval, we can make use of the law of conservation of energy. According to the law, the total mechanical energy of a system remains constant if no external forces (such as air resistance) are acting on it.
The total mechanical energy of the system consists of potential energy (PE) and kinetic energy (KE). At the maximum height, the girl's kinetic energy is zero, and her potential energy is at its maximum. Conversely, at the start of the interval, her kinetic energy is at its maximum, and her potential energy is zero. Therefore, we have the following equation relating kinetic and potential energy:
Initial KE + Initial PE = Final KE + Final PE
At the start of the interval (before leaving the trampoline), the girl's kinetic energy is 480 J, and her potential energy is 0 J. At the end of the interval (maximum height), her kinetic energy is 0 J, and her potential energy is the energy she had initially (480 J). Therefore, we can set up the equation as follows:
480 J + 0 J = 0 J + PE
Simplifying:
480 J = PE
The potential energy (PE) at the maximum height is equal to the work done against gravity. The work done against gravity is equal to the force of gravity (mass times acceleration due to gravity) multiplied by the height. Therefore, we can write:
PE = m * g * h
where m is the mass (27 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
Substituting these values into the equation, we have:
480 J = (27 kg) * (9.8 m/s²) * h
Simplifying further:
480 J = 264.6 kg·m²/s² * h
To isolate h, divide both sides of the equation by 264.6 kg·m²/s²:
h = 480 J / 264.6 kg·m²/s²
Calculating this value:
h ≈ 1.814 m
Therefore, the girl rises approximately 1.814 meters during this interval.