he elastic energy stored in your tendons can contribute up to 35% of your energy needs when running. Sports scientists have studied the change in length of the knee extensor tendon in sprinters and nonathletes. They find that one sprinters’ tendons stretch 46 mm, while a nonathletes’ stretch only 26 mm. The spring constant for the tendon is the same for both groups, 40 N/mm. What is the difference in maximum stored energy between the sprinters and the nonathletes?

To find the difference in maximum stored energy between the sprinters and the nonathletes, we can use the formula for the elastic potential energy stored in a spring (or tendon in this case):

\[
E = \frac{1}{2} k x^2
\]

where:
- \( E \) is the elastic potential energy,
- \( k \) is the spring constant,
- \( x \) is the extension (change in length) of the tendon.

Given dataTo find the difference in maximum stored energy between the sprinters and the nonathletes, we can use the formula for the elastic potential energy stored in a spring (or tendon in this case):

\[
E = \frac{1}{2} k x^2
\]

where:
- \( E \) is the elastic potential energy,
- \( k \) is the spring constant,
- \( x \) is the extension (change in length) of the tendon.

Given 
- Spring constant \( k = 40 \, \text{N/mm} \)
- Extension for the sprinter \( x_s = 46 \, \text{mm} \)
- Extension for the nonathlete \( x_n = 26 \, \text{mm} \)

Let's calculate the elastic potential energy stored in the tendons for both the sprinter and the nonathlete.

### Energy stored in the sprinter's tendon:

\[
E_s = \frac{1}{2} \times 40 \, \text{N/mm} \times (46 \, \text{mm})^2
\]

\[
E_s = 20 \times 2116 \, \text{N·mm}
\]

\[
E_s = 42320 \, \text{N·mm}
\]

### Energy stored in the nonathlete's tendon:

\[
E_n = \frac{1}{2} \times 40 \, \text{N/mm} \times (26 \, \text{mm})^2
\]

\[
E_n = 20 \times 676 \, \text{N·mm}
\]

\[
E_n = 13520 \, \text{N·mm}
\]

### Difference in energy:

\[
\Delta E = E_s - E_n
\]

\[
\Delta E = 42320 \, \text{N·mm} - 13520 \, \text{N·mm}
\]

\[
\Delta E = 28800 \, \text{N·mm}
\]

Since \(1 \, \text{N·mm} = 0.001 \, \text{J}\), converting the energy difference to joules:

\[
\Delta E = 28800 \times 0.001 \, \text{J} = 28.8 \, \text{J}
\]

**The difference in maximum stored energy between the sprinters and the nonathletes is 28.8 joules.**

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How would the energy stored change if the spring constant were different?
2. What other factors besides tendon elasticity might contribute to a sprinter's performance?
3. How does tendon elasticity relate to injury prevention in athletes?
4. What is the significance of elastic energy in other forms of physical activity?
5. Can the spring constant of a tendon change with training or age?

**Tip:** Tendon elasticity and strength play a crucial role in reducing energy expenditure and enhancing performance in various sports.

Discover how to calculate the difference in maximum stored energy between sprinters and nonathletes based on the change in length of knee extensor tendons. Utilize the formula for elastic potential energy and understand the impact of tendon extension differences. Suitable for high school students studying physics or biomechanics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation