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Answer :
To find out how high Avi should reach after compressing the trampoline, we need to follow these steps:
1. Determine Potential Energy in the Trampoline:
When Avi jumps and compresses the trampoline, the trampoline stores energy in the form of potential energy (similar to a spring). The formula to calculate the potential energy stored in a spring is:
[tex]\[
\text{Potential Energy} = \frac{1}{2} \times \text{Spring Constant} \times (\text{Compression})^2
\][/tex]
Here:
- Spring Constant = [tex]\(176,400 \frac{N}{m}\)[/tex]
- Compression = 20 cm = 0.2 m (converted from centimeters to meters)
Plug these values into the formula:
[tex]\[
\text{Potential Energy} = \frac{1}{2} \times 176,400 \times (0.2)^2 = 3,528 \text{ Joules}
\][/tex]
2. Convert Potential Energy to Gravitational Potential Energy:
When the energy stored in the trampoline propels Avi upward, it is converted into gravitational potential energy at the peak of her jump. The gravitational potential energy is given by:
[tex]\[
\text{Gravitational Potential Energy} = \text{Weight} \times g \times \text{Height}
\][/tex]
Where:
- Weight = mass of Avi = 40 kg
- [tex]\(g\)[/tex] = acceleration due to gravity = [tex]\(9.81 \, \frac{m}{s^2}\)[/tex]
The energy stored in the trampoline equals the gravitational potential energy at the height of the jump. Set the two equal to find the height:
[tex]\[
3,528 = 40 \times 9.81 \times \text{Height}
\][/tex]
3. Solve for the Height:
Rearrange the equation to solve for Height:
[tex]\[
\text{Height} = \frac{3,528}{40 \times 9.81}
\][/tex]
Calculate the height:
[tex]\[
\text{Height} = \frac{3,528}{392.4} \approx 8.99 \text{ meters}
\][/tex]
So, Avi should reach a height of approximately 8.99 meters when she jumps on the trampoline.
1. Determine Potential Energy in the Trampoline:
When Avi jumps and compresses the trampoline, the trampoline stores energy in the form of potential energy (similar to a spring). The formula to calculate the potential energy stored in a spring is:
[tex]\[
\text{Potential Energy} = \frac{1}{2} \times \text{Spring Constant} \times (\text{Compression})^2
\][/tex]
Here:
- Spring Constant = [tex]\(176,400 \frac{N}{m}\)[/tex]
- Compression = 20 cm = 0.2 m (converted from centimeters to meters)
Plug these values into the formula:
[tex]\[
\text{Potential Energy} = \frac{1}{2} \times 176,400 \times (0.2)^2 = 3,528 \text{ Joules}
\][/tex]
2. Convert Potential Energy to Gravitational Potential Energy:
When the energy stored in the trampoline propels Avi upward, it is converted into gravitational potential energy at the peak of her jump. The gravitational potential energy is given by:
[tex]\[
\text{Gravitational Potential Energy} = \text{Weight} \times g \times \text{Height}
\][/tex]
Where:
- Weight = mass of Avi = 40 kg
- [tex]\(g\)[/tex] = acceleration due to gravity = [tex]\(9.81 \, \frac{m}{s^2}\)[/tex]
The energy stored in the trampoline equals the gravitational potential energy at the height of the jump. Set the two equal to find the height:
[tex]\[
3,528 = 40 \times 9.81 \times \text{Height}
\][/tex]
3. Solve for the Height:
Rearrange the equation to solve for Height:
[tex]\[
\text{Height} = \frac{3,528}{40 \times 9.81}
\][/tex]
Calculate the height:
[tex]\[
\text{Height} = \frac{3,528}{392.4} \approx 8.99 \text{ meters}
\][/tex]
So, Avi should reach a height of approximately 8.99 meters when she jumps on the trampoline.
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