Let's break down the solution step by step to find out how high Avi, the gymnast, will reach.
### Step 1: Understanding the Data
- Mass of Avi (m): 40 kg
- Spring constant (k): [tex]\(176,400 \, \frac{\text{N}}{\text{m}}\)[/tex]
- Compression of the trampoline (x): 20 cm, which is 0.20 meters (since 1 cm = 0.01 meters)
- Gravitational acceleration (g): [tex]\(9.81 \, \frac{\text{m}}{\text{s}^2}\)[/tex]
### Step 2: Calculate Elastic Potential Energy (EPE)
The formula for the elastic potential energy stored in a compressed spring is:
[tex]\[ \text{EPE} = \frac{1}{2} k x^2 \][/tex]
Plugging in the values:
[tex]\[ \text{EPE} = \frac{1}{2} \times 176,400 \times (0.20)^2 \][/tex]
### Step 3: Use the Elastic Potential Energy to Find the Height Reached
The height reached by Avi can be found by converting the elastic potential energy into gravitational potential energy (GPE), which is given by:
[tex]\[ \text{GPE} = mgh \][/tex]
Since the elastic potential energy will equal the gravitational potential energy at the maximum height, we can write:
[tex]\[ \text{EPE} = mgh \][/tex]
Solving for [tex]\( h \)[/tex] (height reached):
[tex]\[ h = \frac{\text{EPE}}{mg} \][/tex]
From our earlier calculation, the elastic potential energy was found to be:
[tex]\[ \text{EPE} = 3528 \, \text{J} \][/tex]
Now, substitute [tex]\( \text{EPE} \)[/tex], [tex]\( m \)[/tex], and [tex]\( g \)[/tex] into the equation for height:
[tex]\[ h = \frac{3528}{40 \times 9.81} \][/tex]
### Step 4: Simplify the Calculation
Simplifying the above expression:
[tex]\[ h = \frac{3528}{392.4} \approx 8.99 \, \text{meters} \][/tex]
Therefore, Avi should reach approximately 8.99 meters.
