A flea is able to jump straight up about 0.48 m. It has been said that if a flea were as big as a human, it would be able to jump over a 100-story building! When an animal jumps, it converts work done in contracting muscles into gravitational potential energy (with some steps in between). The maximum force exerted by a muscle is proportional to its cross-sectional area, and the work done by the muscle is this force times the length of contraction. If we magnified a flea by a factor of 1035, the cross section of its muscle would increase by 10352 and the length of contraction would increase by 1035. How high would this "super flea" be able to jump? (Don't forget that the mass of the "super flea" increases as well.)

To determine how high this "super flea" could jump after being magnified by a factor of \(10^{35}\), we need to understand how scaling affects different physical quantities. Let's break it down step by step:

### 1. **Scaling of Muscle Force:**
The maximum force exerted by a muscle is proportional to its cross-sectional area. When the flea is magnified by a factor of \(10^{35}\), the cross-sectional area of its muscles increases by the square of this factor:

\[
\text{Cross-sectional area} \propto (10^{35})^2 = 10^{70}
\]

Thus, the force exerted by the super flea's muscles increases by a factor of \(10^{70}\).

### 2. **Scaling of Muscle Contraction:**
The work done by the muscles is the product of the force and the length of the contraction. Since the length of contraction increases by a factor of \(10^{35}\), we multiply the force by this factor to get the work done by the muscles:

\[
\text{Work done} \propto 10^{70} \times 10^{35} = 10^{105}
\]

So, the work done by the super flea's muscles increases by a factor of \(10^{105}\).

### 3. **Scaling of the Flea's Mass:**
The mass of the flea increases in proportion to its volume. Since the flea is magnified by a factor of \(10^{35}\) in each dimension, its volume (and thus its mass) increases by the cube of this factor:

\[
\text{Mass} \propto (10^{35})^3 = 10^{105}
\]

Thus, the mass of the super flea increases by a factor of \(10^{105}\).

### 4. **Height of the Jump:**
The height \(h\) to which an animal can jump is determined by equating the work done by the muscles to the gravitational potential energy:

\[
\text{Work} = \text{Potential Energy} = mgh
\]

Here:
- \(m\) is the mass,
- \(g\) is the gravitational acceleration, and
- \(h\) is the height of the jump.

For the original flea, the work done by the muscles is converted into potential energy, so the flea can jump to a height of 0.48 m.

Now, for the super flea, we know that both the work done and the mass increase by a factor of \(10^{105}\), meaning the ratio of work to mass remains the same:

\[
\frac{\text{Work}_{\text{super flea}}}{\text{Mass}_{\text{super flea}}} = \frac{\text{Work}_{\text{flea}}}{\text{Mass}_{\text{flea}}}
\]

Since this ratio doesn't change, the super flea would jump to the **same height** as the original flea, which is 0.48 m.

### Conclusion:
Even though the super flea is much larger, it would only be able to jump to a height of 0.48 m, just like the original flea. This is because both the work done by the muscles and the mass scale equally with the size of the flea, leaving the jump height unchanged.

Would you like further clarification or details on any step? Here are some related questions you might explore:

1. How does the scaling law for muscle force apply to different animals?
2. What other factors besides muscle force affect an animal's jump height?
3. How would air resistance impact the jump height of larger creatures?
4. Could a human-sized flea survive under Earth’s gravity?
5. How would the strength of materials limit the size of living organisms?

**Tip:** When scaling physical systems, it's important to understand which factors (like force, mass, and energy) scale linearly, quadratically, or cubically with size.

Explore how scaling laws affect physical quantities like muscle force, work, and mass in this detailed problem. Learn why a flea scaled by 10^35 would still jump to a height of only 0.48 meters due to proportional relationships in force and mass. Suitable for advanced high school or college students.

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