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 150 J from 430 J. How high does she rise during this interval? Neglect air resistance.
To solve this problem, we can use the principle of conservation of energy. The total mechanical energy of the girl-trampoline system remains constant throughout the motion, neglecting air resistance.
The mechanical energy is the sum of potential energy and kinetic energy:
Total Mechanical Energy = Potential Energy + Kinetic Energy
We can assume that the girl starts from rest at the surface of the trampoline, so initially, her potential energy is zero, and her kinetic energy is equal to the total mechanical energy.
At the highest point of her motion, all her initial kinetic energy is converted to potential energy, so her potential energy is equal to the total mechanical energy.
Given that her initial kinetic energy is 430 J and her final kinetic energy is 150 J, the change in kinetic energy is:
Change in Kinetic Energy = Final Kinetic Energy - Initial Kinetic Energy
Change in Kinetic Energy = 150 J - 430 J
Change in Kinetic Energy = -280 J
Since the mechanical energy is conserved, the change in kinetic energy is equal to the change in potential energy:
Change in Potential Energy = -280 J
At the highest point of her motion, her potential energy is equal to her final kinetic energy:
Potential Energy = Final Kinetic Energy
Potential Energy = 150 J
Therefore, her potential energy increases by 280 J from zero to 150 J. This increase in potential energy represents the work done against gravity to lift the girl.
The potential energy equation is given by:
Potential Energy = m * g * h
where m is the mass of the girl (27 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the girl's rise.
Rearranging the equation to solve for h:
h = Potential Energy / (m * g)
h = (280 J) / (27 kg * 9.8 m/s^2)
h ≈ 1.1 meters
Therefore, the girl rises approximately 1.1 meters during this interval.
To calculate the height to which the girl rises, we need to use the principle of conservation of mechanical energy. The mechanical energy of the system (girl + trampoline) is conserved in the absence of external forces like air resistance.
The mechanical energy of the system is the sum of the potential energy (PE) and the kinetic energy (KE) of the girl.
At the highest point of her trajectory, when the girl is momentarily at rest before descending, all of her initial kinetic energy will be converted into potential energy.
So, we have the equation:
PE(initial) + KE(initial) = PE(final) + KE(final)
Now, let's calculate the potential energy at the initial and final points:
PE(initial) = m*g*h (where m is the mass of the girl, g is the acceleration due to gravity, and h is the height)
KE(initial) = 430 J (given in the question)
PE(final) = 0 J (at the highest point, the potential energy is maximum and the kinetic energy is zero)
KE(final) = 150 J (given in the question)
We can plug in the given values into the equation and solve for h:
m*g*h + KE(initial) = PE(final) + KE(final)
27 kg * 9.8 m/s^2 * h + 430 J = 0 J + 150 J
264.6 h + 430 = 150
264.6 h = -280
h = -280 / 264.6
h ≈ -1.06 meters
The negative sign indicates a downward displacement. However, since the question asks for the height above the trampoline surface, we take the magnitude of the height:
h ≈ 1.06 meters
Therefore, the girl rises approximately 1.06 meters during this interval.