To analyze the scenario involving Cesar and Jake racing, we will apply Newton's second law of motion, which is expressed as:
\[ F = m \cdot a \]
where:
- \( F \) is the force applied,
- \( m \) is the mass,
- \( a \) is the acceleration.
Given:
- Cesar's mass \( m_C = 600 , \text{N} \)
- Jake's mass \( m_J = 900 , \text{N} \)
To clarify: the "mass" given in newtons is actually the weight of each boy. To find the mass, we will use the equation:
\[ m = \frac{weight}{g} \]
where \( g \) (acceleration due to gravity) is approximately \( 9.81 , \text{m/s}^2 \).
Calculating the masses:
- Cesar:
\[ m_C = \frac{600 , \text{N}}{9.81 , \text{m/s}^2} \approx 61.15 , \text{kg} \]
- Jake:
\[ m_J = \frac{900 , \text{N}}{9.81 , \text{m/s}^2} \approx 91.83 , \text{kg} \]
Factors that Affect the Outcome of the Race
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Acceleration: The ability of each boy to accelerate will depend on the net force they can exert and their mass. More force will lead to greater acceleration, which is crucial for winning a race.
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Force Exerted: The amount of force each boy can exert will also suspend the outcome. If both boys apply forces, we need to find the equations for acceleration based on the force they can exert.
Scenario Where Cesar Wins
Suppose Cesar is able to exert a force of \( 800 , \text{N} \) while Jake can only exert a force of \( 900 , \text{N} \). We calculate their accelerations:
\[ a_C = \frac{F_C}{m_C} = \frac{800 , \text{N}}{61.15 , \text{kg}} \approx 13.09 , \text{m/s}^2 \]
\[ a_J = \frac{F_J}{m_J} = \frac{900 , \text{N}}{91.83 , \text{kg}} \approx 9.80 , \text{m/s}^2 \]
In this scenario, since \( a_C > a_J \), Cesar will win the race because his acceleration is greater than Jake's, despite Jake having a greater weight.
Scenario Where Jake Wins
Now suppose Jake can exert a force of \( 1500 , \text{N} \) while Cesar exerts only \( 700 , \text{N} \):
\[ a_C = \frac{F_C}{m_C} = \frac{700 , \text{N}}{61.15 , \text{kg}} \approx 11.43 , \text{m/s}^2 \]
\[ a_J = \frac{F_J}{m_J} = \frac{1500 , \text{N}}{91.83 , \text{kg}} \approx 16.33 , \text{m/s}^2 \]
In this scenario, since \( a_J > a_C \), Jake will win the race due to his larger acceleration from the greater force he can exert.
Conclusion
The key factors that influence the outcome of the race are the forces exerted by each boy and their respective masses. The scenario can vary based on the capabilities each boy possesses in terms of exerting force, allowing either boy to win under different conditions.
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