A 26-kg girl is bouncing on a trampoline. During a certain interval after leaving the surface of the trampoline, her kinetic energy decreases to 220 J from 500 J. How high does she rise during this interval? Neglect air resistance.
Well, let's bounce into some calculations, shall we? Since the girl's kinetic energy decreases, we can assume that some of it is converted into potential energy as she rises. So, we can use the principle of conservation of energy.
Her initial kinetic energy, Ki, is 500 J, and her final kinetic energy, Kf, is 220 J. The difference between the two is the potential energy gained, which we can call PE:
PE = Ki - Kf
PE = 500 J - 220 J
PE = 280 J
But potential energy can also be calculated as the product of mass (m), gravitational acceleration (g), and height (h):
PE = mgh
Since g is approximately 9.8 m/s², we can rearrange the equation to solve for h:
h = PE / (mg)
h = 280 J / (26 kg * 9.8 m/s²)
h ≈ 1.10 meters
So, during this interval, the girl rises to a height of approximately 1.10 meters. Keep bouncing with that math knowledge!
To determine the height the girl rises during this interval, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system, which includes the sum of potential energy and kinetic energy, remains constant as long as no external forces are acting on the system.
In this case, the girl's potential energy at the highest point of her bounce would be equal to her initial kinetic energy. The difference in kinetic energy (ΔKE) is used to calculate the change in potential energy (ΔPE), which can then be converted to the change in height (Δh).
Given:
Mass of the girl, m = 26 kg
Initial kinetic energy, KE_initial = 500 J
Final kinetic energy, KE_final = 220 J
Step 1: Calculate the change in kinetic energy
ΔKE = KE_final - KE_initial
ΔKE = 220 J - 500 J
ΔKE = -280 J
Step 2: Calculate the change in potential energy
ΔPE = -ΔKE
ΔPE = 280 J (since ΔKE is negative)
Step 3: Calculate the change in height (Δh)
ΔPE = m * g * Δh
280 J = 26 kg * 9.8 m/s^2 * Δh
Solving for Δh:
Δh = 280 J / (26 kg * 9.8 m/s^2)
Δh ≈ 1.11 m
Therefore, during this interval, the girl rises approximately 1.11 meters.
To find the height the girl rises during this interval, we can make use of the principle of conservation of energy. According to this principle, the total mechanical energy of an object remains constant in the absence of external forces such as air resistance.
The total mechanical energy of the girl-trampoline system is the sum of her kinetic energy (KE) and her potential energy (PE).
Given that the initial kinetic energy (KEi) is 500 J, and the final kinetic energy (KEf) is 220 J, we can write the equation:
KEi + PEi = KEf + PEf
Since she leaves the surface of the trampoline, her potential energy at the beginning (PEi) is zero. Therefore, the equation simplifies to:
KEi = KEf + PEf
Rearranging the equation to solve for the potential energy (PEf):
PEf = KEi - KEf
PEf = 500 J - 220 J
PEf = 280 J
Now, we can use the formula for potential energy in the context of gravitational potential energy:
PE = mgh
Where:
PE is the potential energy,
m is the mass of the object,
g is the acceleration due to gravity (9.8 m/s^2),
and h is the height.
Rearranging the equation to solve for the height:
h = PE / (mg)
Substituting the values:
h = 280 J / (26 kg × 9.8 m/s^2)
h ≈ 1.08 meters
Therefore, the girl rises approximately 1.08 meters during this interval.