A 45-kg girl is bouncing on a trampoline. During a certain interval after leaving the surface of the trampoline, her kinetic energy decreases to 210 J from 400 J. How high does she rise during this interval? Neglect air resistance.
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To determine the height the girl rises during the interval, we need to find the potential energy associated with her change in kinetic energy. According to the law of conservation of energy, the sum of kinetic energy and potential energy remains constant in the absence of external forces such as air resistance.
We are given that the girl's kinetic energy decreases from 400 J to 210 J. Therefore, the change in her kinetic energy, ΔKE, is given by:
ΔKE = Final KE - Initial KE
ΔKE = 210 J - 400 J
ΔKE = -190 J
The negative sign indicates a decrease in kinetic energy.
Since the girl loses kinetic energy, an equal amount of potential energy is gained. Therefore, the change in potential energy, ΔPE, is equal to -ΔKE.
ΔPE = -(-190 J)
ΔPE = 190 J
Now, we can calculate the height the girl rises using the formula for gravitational potential energy:
ΔPE = mgh
Where:
ΔPE is the change in potential energy (190 J)
m is the mass (45 kg)
g is the acceleration due to gravity (9.8 m/s²)
h is the height
Rearranging the formula, we can solve for h:
h = ΔPE / (mg)
h = 190 J / (45 kg * 9.8 m/s²)
h ≈ 0.433 m
Therefore, the girl rises approximately 0.433 meters during this interval.