First, let's understand the problem:
- Avi weighs 40 kg.
- The spring constant of the trampoline is [tex]\(176,400 \frac{N}{m}\)[/tex].
- Avi compresses the trampoline by [tex]\(20 \text{ cm}\)[/tex], which is [tex]\(0.20 \text{ meters}\)[/tex].
To find out how high Avi should reach, we follow these steps:
1. Calculate the Potential Energy (PE) stored in the compressed trampoline:
The potential energy stored in a compressed spring is given by the formula:
[tex]\[
PE = \frac{1}{2} k x^2
\][/tex]
where [tex]\(k\)[/tex] is the spring constant and [tex]\(x\)[/tex] is the compression distance.
Plugging in the values:
[tex]\[
PE = \frac{1}{2} \times 176,400 \frac{N}{m} \times (0.20 \text{ m})^2
\][/tex]
[tex]\[
PE = \frac{1}{2} \times 176,400 \times 0.04
\][/tex]
[tex]\[
PE = 3,528 \text{ Joules}
\][/tex]
2. Convert Avi's weight to force:
The force due to gravity on Avi can be calculated as:
[tex]\[
F = m \times g
\][/tex]
where [tex]\(m\)[/tex] is Avi's mass (40 kg) and [tex]\(g\)[/tex] is the acceleration due to gravity, which is [tex]\(9.81 \frac{m}{s^2}\)[/tex].
[tex]\[
F = 40 \text{ kg} \times 9.81 \frac{m}{s^2}
\][/tex]
[tex]\[
F = 392.4 \text{ Newtons}
\][/tex]
3. Determine the height (h) she should reach:
The potential energy converted to gravitational potential energy when she reaches a height [tex]\(h\)[/tex] is given by:
[tex]\[
PE = m \times g \times h
\][/tex]
We know the potential energy (3,528 Joules) and the force (392.4 Newtons), hence:
[tex]\[
3,528 = 392.4 \times h
\][/tex]
Solving for [tex]\(h\)[/tex]:
[tex]\[
h = \frac{3,528}{392.4}
\][/tex]
[tex]\[
h \approx 8.99 \text{ meters}
\][/tex]
Therefore, Avi should reach approximately [tex]\(8.99 \text{ meters}\)[/tex] high.
