A 42 kg girl is bouncing on a trampoline. During a certain interval after she leaves the

surface of the trampoline, her kinetic energy decreases to 250 J from 520 J. How high
does she rise during this interval?

KE+PE = 520 J.

250 + PE = 520
PE = 520-250 = 270 J.
mg*h = 270
h = 270/m*g = 270/(42*9.8) = 0.656 m.

To determine the height the girl rises during this interval, we can use the principle of conservation of energy. The initial kinetic energy (KE1) of the girl is equal to the final potential energy (PE2) when she reaches her maximum height.

Step 1: Determine the initial potential energy (PE1) of the girl.
Given:
Mass (m) = 42 kg
Height (h1) = ?

The formula for potential energy is:
PE = m * g * h

Where:
m = mass of the girl
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height above the reference point

Substituting the given values into the formula:
PE1 = m * g * h1

Step 2: Determine the final height (h2) of the girl.
The final potential energy (PE2) is given as 250 J.
Hence,
PE2 = m * g * h2

Step 3: Calculate the change in kinetic energy (ΔKE).
The change in kinetic energy is given by:
ΔKE = KE2 - KE1

Given:
KE2 = 250 J
KE1 = 520 J

ΔKE = 250 J - 520 J
= -270 J

The change in kinetic energy (ΔKE) is negative because the kinetic energy decreases.

Step 4: Apply the principle of conservation of energy to determine the height (h2).
Since the initial kinetic energy (KE1) is equal to the final potential energy (PE2) when the maximum height is reached:
KE1 = PE2

Substituting the respective formulas:
m * v^2 /2 = m * g * h2

Where:
m = mass of the girl
v = velocity of the girl
g = acceleration due to gravity
h2 = maximum height reached

Since the velocity (v) is cancelled out, the formula can be simplified to:
v^2 /2 = g * h2

Step 5: Solve for the maximum height (h2).
Rearranging the formula for h2 gives:
h2 = v^2 /(2 * g)

Substituting the given values into the formula:
h2 = (-270 J) / (2 * 9.8 m/s^2)
= (-270 J) / 19.6 m/s^2
≈ -13.8 m

The negative sign means that the girl does not actually rise to a height of -13.8 m. It indicates that during the interval, the girl is descending from her maximum height.

Therefore, during this interval, the girl descends approximately 13.8 meters.

To find the height the girl rises during this interval, we need to use the concept of conservation of energy.

The total energy of the girl when she is on the trampoline is the sum of her potential energy and kinetic energy. When she leaves the surface and rises up, her potential energy increases, while her kinetic energy decreases.

The total energy of an object can be calculated using the formula:

Total Energy = Potential Energy + Kinetic Energy

We can set up an equation using this principle:

Total Energy before = Total Energy after

Given that her kinetic energy before is 520 J and after is 250 J, we have:

Potential Energy before + Kinetic Energy before = Potential Energy after + Kinetic Energy after

Let's denote the height the girl rises as h. The potential energy of an object near the surface of the Earth is given by the formula:

Potential Energy = mass * gravitational acceleration * height

Given that the mass of the girl is 42 kg and the gravitational acceleration is approximately 9.8 m/s², we can rewrite the equation:

42 * 9.8 * h + 520 = 42 * 9.8 * 0 + 250

Simplifying the equation:

411.6h + 520 = 250

Subtracting 520 from both sides:

411.6h = 250 - 520

411.6h = -270

Dividing both sides by 411.6:

h = -270 / 411.6

Calculating the value:

h ≈ -0.656 meters

The negative sign indicates that the girl falls below the surface of the trampoline during this interval. Therefore, she does not rise during this interval.