Question
Why was Bolt's mass a mathematical disadvantage to reaching high acceleration? Use the drawing space to provide evidence for your response using examples of values and the most appropriate formula. Provide your explanation in the text field provided.
Extra Help
Consider these questions when forming your response:
- How can the formula F = ma be rearranged?
- Which arrangement of the formula would best show how mass affects acceleration?
- What values can you use for mass to show how a larger mass affects acceleration?
Extra Help
Consider these questions when forming your response:
- How can the formula F = ma be rearranged?
- Which arrangement of the formula would best show how mass affects acceleration?
- What values can you use for mass to show how a larger mass affects acceleration?
Answers
GPT-4o mini
To understand why Usain Bolt's mass could be a disadvantage in reaching high accelerations, we can apply Newton's second law of motion, which is expressed as:
\[ F = ma \]
Where:
- \( F \) is the net force applied,
- \( m \) is the mass of the object,
- \( a \) is the acceleration.
We can rearrange this formula to find acceleration:
\[ a = \frac{F}{m} \]
From this rearrangement, it’s clear that acceleration (\( a \)) is inversely proportional to mass (\( m \)) when force (\( F \)) is kept constant. This means that as mass increases, acceleration decreases if the force applied does not change.
### Example Values
Let’s consider two hypothetical athletes with different masses but the same force exerted during a sprint.
1. **Athlete A (lower mass)**:
- Mass (\( m \)): 70 kg
- Force (\( F \)): 900 N (a strong push-off from the blocks)
- Acceleration (\( a \)):
\[
a = \frac{F}{m} = \frac{900\,N}{70\,kg} \approx 12.86\,m/s^2
\]
2. **Athlete B (higher mass)**:
- Mass (\( m \)): 95 kg (e.g., Usain Bolt)
- Force (\( F \)): 900 N (same push-off)
- Acceleration (\( a \)):
\[
a = \frac{F}{m} = \frac{900\,N}{95\,kg} \approx 9.47\,m/s^2
\]
### Analysis
From our calculations:
- Athlete A achieves an acceleration of approximately \( 12.86\,m/s^2 \).
- Athlete B achieves an acceleration of approximately \( 9.47\,m/s^2 \).
This demonstrates that with a higher mass (like Bolt’s), the same amount of force results in less acceleration. Therefore, despite Bolt being an exceptionally fast sprinter with significant strength, his relatively higher mass caused him to have a lower rate of acceleration compared to lighter athletes.
### Conclusion
In sprinting, particularly during the initial phase of a race, achieving high acceleration is crucial for maximizing speed. The greater the mass, the less acceleration can be achieved for a given force, which can hinder the overall performance in a race, especially when explosive starts are critical. Consequently, Bolt's mass could be viewed as a disadvantage in reaching high acceleration quickly, as shown by the mathematical relationship derived from \( F = ma \).
\[ F = ma \]
Where:
- \( F \) is the net force applied,
- \( m \) is the mass of the object,
- \( a \) is the acceleration.
We can rearrange this formula to find acceleration:
\[ a = \frac{F}{m} \]
From this rearrangement, it’s clear that acceleration (\( a \)) is inversely proportional to mass (\( m \)) when force (\( F \)) is kept constant. This means that as mass increases, acceleration decreases if the force applied does not change.
### Example Values
Let’s consider two hypothetical athletes with different masses but the same force exerted during a sprint.
1. **Athlete A (lower mass)**:
- Mass (\( m \)): 70 kg
- Force (\( F \)): 900 N (a strong push-off from the blocks)
- Acceleration (\( a \)):
\[
a = \frac{F}{m} = \frac{900\,N}{70\,kg} \approx 12.86\,m/s^2
\]
2. **Athlete B (higher mass)**:
- Mass (\( m \)): 95 kg (e.g., Usain Bolt)
- Force (\( F \)): 900 N (same push-off)
- Acceleration (\( a \)):
\[
a = \frac{F}{m} = \frac{900\,N}{95\,kg} \approx 9.47\,m/s^2
\]
### Analysis
From our calculations:
- Athlete A achieves an acceleration of approximately \( 12.86\,m/s^2 \).
- Athlete B achieves an acceleration of approximately \( 9.47\,m/s^2 \).
This demonstrates that with a higher mass (like Bolt’s), the same amount of force results in less acceleration. Therefore, despite Bolt being an exceptionally fast sprinter with significant strength, his relatively higher mass caused him to have a lower rate of acceleration compared to lighter athletes.
### Conclusion
In sprinting, particularly during the initial phase of a race, achieving high acceleration is crucial for maximizing speed. The greater the mass, the less acceleration can be achieved for a given force, which can hinder the overall performance in a race, especially when explosive starts are critical. Consequently, Bolt's mass could be viewed as a disadvantage in reaching high acceleration quickly, as shown by the mathematical relationship derived from \( F = ma \).