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Answer :
To solve this problem, we need to calculate how high Avi reaches when she jumps on the trampoline. We can do this by using the principles of energy conservation.
1. Understanding the Problem:
- Avi weighs 40 kg.
- The trampoline, when compressed 20 cm (which is 0.2 meters), has a spring constant of 176,400 N/m.
- We want to find the maximum height Avi reaches from this jump.
2. Energy Conservation:
- The energy stored in the trampoline as it is compressed is potential energy from the spring.
- When Avi jumps, this potential energy is converted into gravitational potential energy at the highest point of her jump.
3. Calculate Potential Energy from the Spring:
- The formula for the potential energy stored in a spring is:
[tex]\[
\text{PE}_{\text{spring}} = \frac{1}{2} \times \text{spring constant} \times (\text{compression distance})^2
\][/tex]
- Plug in the values:
[tex]\[
\text{PE}_{\text{spring}} = \frac{1}{2} \times 176,400 \times (0.2)^2
\][/tex]
4. Calculate Gravitational Potential Energy:
- At the highest point, all the spring's potential energy is converted to gravitational potential energy:
[tex]\[
\text{PE}_{\text{gravitational}} = \text{mass} \times g \times \text{height}
\][/tex]
- Here, [tex]\( g \)[/tex] is the acceleration due to gravity, which is approximately 9.81 m/s².
5. Set the Energies Equal to Solve for Height:
- Since the energy is conserved, set spring potential energy equal to gravitational potential energy:
[tex]\[
\frac{1}{2} \times 176,400 \times (0.2)^2 = 40 \times 9.81 \times \text{height}
\][/tex]
- Solve for [tex]\(\text{height}\)[/tex]:
[tex]\[
\text{height} = \frac{\frac{1}{2} \times 176,400 \times (0.2)^2}{40 \times 9.81}
\][/tex]
6. Result:
- When we perform these calculations, we find the potential energy stored in the trampoline is approximately 3528 J (joules).
- The height Avi reaches is approximately 8.99 meters.
Therefore, Avi should reach a height of about 8.99 meters when she jumps on the trampoline.
1. Understanding the Problem:
- Avi weighs 40 kg.
- The trampoline, when compressed 20 cm (which is 0.2 meters), has a spring constant of 176,400 N/m.
- We want to find the maximum height Avi reaches from this jump.
2. Energy Conservation:
- The energy stored in the trampoline as it is compressed is potential energy from the spring.
- When Avi jumps, this potential energy is converted into gravitational potential energy at the highest point of her jump.
3. Calculate Potential Energy from the Spring:
- The formula for the potential energy stored in a spring is:
[tex]\[
\text{PE}_{\text{spring}} = \frac{1}{2} \times \text{spring constant} \times (\text{compression distance})^2
\][/tex]
- Plug in the values:
[tex]\[
\text{PE}_{\text{spring}} = \frac{1}{2} \times 176,400 \times (0.2)^2
\][/tex]
4. Calculate Gravitational Potential Energy:
- At the highest point, all the spring's potential energy is converted to gravitational potential energy:
[tex]\[
\text{PE}_{\text{gravitational}} = \text{mass} \times g \times \text{height}
\][/tex]
- Here, [tex]\( g \)[/tex] is the acceleration due to gravity, which is approximately 9.81 m/s².
5. Set the Energies Equal to Solve for Height:
- Since the energy is conserved, set spring potential energy equal to gravitational potential energy:
[tex]\[
\frac{1}{2} \times 176,400 \times (0.2)^2 = 40 \times 9.81 \times \text{height}
\][/tex]
- Solve for [tex]\(\text{height}\)[/tex]:
[tex]\[
\text{height} = \frac{\frac{1}{2} \times 176,400 \times (0.2)^2}{40 \times 9.81}
\][/tex]
6. Result:
- When we perform these calculations, we find the potential energy stored in the trampoline is approximately 3528 J (joules).
- The height Avi reaches is approximately 8.99 meters.
Therefore, Avi should reach a height of about 8.99 meters when she jumps on the trampoline.
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Rewritten by : Barada
