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Answer :
Sure, let's break down the solution step-by-step for this problem.
1. Understanding the Problem:
- Avi weighs 40 kg.
- The trampoline has a spring constant of [tex]\(176,400 \, \text{N/m}\)[/tex].
- Avi compresses the trampoline by 20 cm (which is 0.2 meters).
- We want to find out how high she will be able to reach after being launched into the air by the trampoline.
2. Convert Weight to Force:
- The force due to Avi's weight can be calculated using her mass.
- The formula to convert mass to force is [tex]\(F = m \times g\)[/tex], where [tex]\(g\)[/tex] is the acceleration due to gravity (approximately [tex]\(9.8 \, \text{m/s}^2\)[/tex]).
- So, the force of her weight is [tex]\(40 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 392 \, \text{N}\)[/tex].
3. Calculate Elastic Potential Energy:
- When avi compresses the trampoline, energy is stored as elastic potential energy. The formula for elastic potential energy in a spring is:
[tex]\[
E = \frac{1}{2} k x^2
\][/tex]
where [tex]\(k\)[/tex] is the spring constant, and [tex]\(x\)[/tex] is the compression distance.
- Substituting the given values, [tex]\(k = 176,400 \, \text{N/m}\)[/tex] and [tex]\(x = 0.2 \, \text{m}\)[/tex], we calculate:
[tex]\[
E = \frac{1}{2} \times 176,400 \, \text{N/m} \times (0.2 \, \text{m})^2 = 3528 \, \text{Joules}
\][/tex]
4. Calculate the Height Reached:
- Once Avi is launched into the air, this elastic potential energy will convert into gravitational potential energy, which is given by:
[tex]\[
E = m \times g \times h
\][/tex]
- Here, we need to find the height [tex]\(h\)[/tex]. Rearranging the formula gives:
[tex]\[
h = \frac{E}{m \times g}
\][/tex]
- Using [tex]\(E = 3528 \, \text{Joules}\)[/tex], [tex]\(m = 40 \, \text{kg}\)[/tex], and [tex]\(g = 9.8 \, \text{m/s}^2\)[/tex], we find:
[tex]\[
h = \frac{3528}{40 \times 9.8} = 9 \, \text{meters}
\][/tex]
So, Avi should reach a height of approximately 9 meters after being launched by the trampoline.
1. Understanding the Problem:
- Avi weighs 40 kg.
- The trampoline has a spring constant of [tex]\(176,400 \, \text{N/m}\)[/tex].
- Avi compresses the trampoline by 20 cm (which is 0.2 meters).
- We want to find out how high she will be able to reach after being launched into the air by the trampoline.
2. Convert Weight to Force:
- The force due to Avi's weight can be calculated using her mass.
- The formula to convert mass to force is [tex]\(F = m \times g\)[/tex], where [tex]\(g\)[/tex] is the acceleration due to gravity (approximately [tex]\(9.8 \, \text{m/s}^2\)[/tex]).
- So, the force of her weight is [tex]\(40 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 392 \, \text{N}\)[/tex].
3. Calculate Elastic Potential Energy:
- When avi compresses the trampoline, energy is stored as elastic potential energy. The formula for elastic potential energy in a spring is:
[tex]\[
E = \frac{1}{2} k x^2
\][/tex]
where [tex]\(k\)[/tex] is the spring constant, and [tex]\(x\)[/tex] is the compression distance.
- Substituting the given values, [tex]\(k = 176,400 \, \text{N/m}\)[/tex] and [tex]\(x = 0.2 \, \text{m}\)[/tex], we calculate:
[tex]\[
E = \frac{1}{2} \times 176,400 \, \text{N/m} \times (0.2 \, \text{m})^2 = 3528 \, \text{Joules}
\][/tex]
4. Calculate the Height Reached:
- Once Avi is launched into the air, this elastic potential energy will convert into gravitational potential energy, which is given by:
[tex]\[
E = m \times g \times h
\][/tex]
- Here, we need to find the height [tex]\(h\)[/tex]. Rearranging the formula gives:
[tex]\[
h = \frac{E}{m \times g}
\][/tex]
- Using [tex]\(E = 3528 \, \text{Joules}\)[/tex], [tex]\(m = 40 \, \text{kg}\)[/tex], and [tex]\(g = 9.8 \, \text{m/s}^2\)[/tex], we find:
[tex]\[
h = \frac{3528}{40 \times 9.8} = 9 \, \text{meters}
\][/tex]
So, Avi should reach a height of approximately 9 meters after being launched by the trampoline.
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