Answer :
To solve this problem of how high Avi should reach when jumping on a trampoline, we need to use the concepts of energy conversion. Let's break it down step-by-step:
1. Understand the Problem:
- Avi's weight: 40 kg
- Spring constant of the trampoline: 176,400 N/m
- Compression distance of the trampoline: 20 cm (convert this to meters: 0.2 m)
- We want to find the maximum height Avi reaches after the jump.
2. Potential Energy Stored in the Trampoline:
- When Avi compresses the trampoline, she stores potential energy in it.
- The potential energy (PE) stored in a spring is calculated using the formula:
[tex]\[
\text{Potential Energy} = \frac{1}{2} \times \text{Spring Constant} \times (\text{Compression Distance})^2
\][/tex]
- Substituting the values we have:
[tex]\[
\text{Potential Energy} = \frac{1}{2} \times 176,400 \times (0.2)^2 = 3528 \text{ Joules}
\][/tex]
3. Conversion to Gravitational Potential Energy:
- At the maximum height, all of the elastic potential energy is converted into gravitational potential energy.
- Gravitational potential energy (GPE) can be calculated using:
[tex]\[
\text{Gravitational Potential Energy} = \text{mass} \times \text{gravitational acceleration} \times \text{height}
\][/tex]
- The gravitational acceleration (g) is approximately 9.81 m/s².
- Setting the potential energy equal to the gravitational potential energy, solve for height (h):
[tex]\[
3528 = 40 \times 9.81 \times h
\][/tex]
4. Calculate the Maximum Height:
- Rearrange the equation to solve for height (h):
[tex]\[
h = \frac{3528}{40 \times 9.81} \approx 8.99 \text{ meters}
\][/tex]
Therefore, Avi should reach a maximum height of approximately 8.99 meters. This is significantly higher than the given choice of 1.8 meters, indicating an alternative evaluation or context might be needed if this seems unexpected.
1. Understand the Problem:
- Avi's weight: 40 kg
- Spring constant of the trampoline: 176,400 N/m
- Compression distance of the trampoline: 20 cm (convert this to meters: 0.2 m)
- We want to find the maximum height Avi reaches after the jump.
2. Potential Energy Stored in the Trampoline:
- When Avi compresses the trampoline, she stores potential energy in it.
- The potential energy (PE) stored in a spring is calculated using the formula:
[tex]\[
\text{Potential Energy} = \frac{1}{2} \times \text{Spring Constant} \times (\text{Compression Distance})^2
\][/tex]
- Substituting the values we have:
[tex]\[
\text{Potential Energy} = \frac{1}{2} \times 176,400 \times (0.2)^2 = 3528 \text{ Joules}
\][/tex]
3. Conversion to Gravitational Potential Energy:
- At the maximum height, all of the elastic potential energy is converted into gravitational potential energy.
- Gravitational potential energy (GPE) can be calculated using:
[tex]\[
\text{Gravitational Potential Energy} = \text{mass} \times \text{gravitational acceleration} \times \text{height}
\][/tex]
- The gravitational acceleration (g) is approximately 9.81 m/s².
- Setting the potential energy equal to the gravitational potential energy, solve for height (h):
[tex]\[
3528 = 40 \times 9.81 \times h
\][/tex]
4. Calculate the Maximum Height:
- Rearrange the equation to solve for height (h):
[tex]\[
h = \frac{3528}{40 \times 9.81} \approx 8.99 \text{ meters}
\][/tex]
Therefore, Avi should reach a maximum height of approximately 8.99 meters. This is significantly higher than the given choice of 1.8 meters, indicating an alternative evaluation or context might be needed if this seems unexpected.
