A 30-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.
m g h = (435-205)
h = (435-25)/( 30*9.81)
To find the height the girl rises during the interval, we can use the principle of conservation of energy. The total mechanical energy of the system (girl and trampoline) remains constant in the absence of external forces.
The initial total mechanical energy is given by the sum of the girl's kinetic energy and potential energy, while the final total mechanical energy is given only by the girl's potential energy.
Since the girl loses kinetic energy, it is converted into potential energy as she rises. Therefore, we can equate the difference in kinetic energy to the change in potential energy.
ΔKE = PE_f - PE_i
The change in kinetic energy (ΔKE) is the final kinetic energy minus the initial kinetic energy:
ΔKE = 205 J - 435 J = -230 J
The negative sign indicates the decrease in kinetic energy.
The change in potential energy (ΔPE) is equal to the negative of the change in kinetic energy (since one energy is decreasing and the other is increasing):
ΔPE = -ΔKE = 230 J
The change in potential energy is given by the formula:
ΔPE = m * g * Δh
Where:
m is the mass of the girl (30 kg)
g is the acceleration due to gravity (9.8 m/s^2)
Δh is the change in height
Plugging in the values:
230 J = 30 kg * 9.8 m/s^2 * Δh
Simplifying the equation, we can solve for Δh:
Δh = 230 J / (30 kg * 9.8 m/s^2)
Calculating the result:
Δh ≈ 0.78 m
Therefore, the girl rises approximately 0.78 meters during the interval.