Bicyclists in the Tour de France do enormous amounts of work during a race. For example, the average power per kilogram generated by Lance Armstrong (m = 75.0 kg) is 6.50 W per kilogram of his body mass. (a) How much work does he do during a 123-km race in which his average speed is 11.0 m/s? (b) Often, the work done is expressed in nutritional Calories rather than in joules. Express the work done in part (a) in terms of nutritional Calories, noting that 1 joule = 2.389 x 10-4 nutritional Calories.
To calculate the work done by Lance Armstrong during the race, we can use the formula:
Work = Force × Distance
But in this case, we need to find the force exerted by Lance Armstrong. The force can be calculated using the average power per kilogram as:
Force = Power per kilogram × Mass
(a) First, let's calculate the force exerted by Lance Armstrong:
Force = 6.50 W/kg × 75.0 kg
Force = 487.5 N
Now, we can calculate the work done by multiplying the force by the distance:
Work = Force × Distance
Work = 487.5 N × 123,000 m
Work = 60,037,500 J
(b) To express the work done in nutritional Calories, we can use the conversion factor given:
1 joule = 2.389 × 10^-4 nutritional Calories
Let's convert the work done to nutritional Calories:
Work (in nutritional Calories) = Work (in joules) × (1 joule / 2.389 × 10^-4 nutritional Calories)
Work (in nutritional Calories) = 60,037,500 J × (1 joule / 2.389 × 10^-4 nutritional Calories)
Work (in nutritional Calories) ≈ 2.511 × 10^8 nutritional Calories
Therefore, the work done by Lance Armstrong during the 123-km race is approximately 2.511 × 10^8 nutritional Calories.
To find the work done by Lance Armstrong during the race, we can use the formula for work which is given by:
Work = Force x Distance
First, let's calculate the force exerted by Lance Armstrong using his power-to-mass ratio:
Power-to-mass ratio = Power / Mass
Given that the power-to-mass ratio is 6.50 W/kg and Lance Armstrong's mass is 75.0 kg, we can calculate the power as:
Power = Power-to-mass ratio x Mass
= 6.50 W/kg x 75.0 kg
= 487.5 W
Next, we can calculate the force by using Newton's second law of motion which states that Force = Mass x Acceleration. Since the race is on a flat surface, the acceleration can be assumed to be the average speed divided by the duration of the race.
Acceleration = Average Speed / Time
Given that the average speed is 11.0 m/s and the distance of the race is 123 km (or 123,000 m), we can calculate the time as:
Time = Distance / Average Speed
= 123,000 m / 11.0 m/s
= 11,181.82 s
Acceleration = 11.0 m/s / 11,181.82 s
= 9.845 x 10^-4 m/s^2
Lastly, we can calculate the force by multiplying the mass with the acceleration:
Force = Mass x Acceleration
= 75.0 kg x 9.845 x 10^-4 m/s^2
= 7.384 x 10^-2 N
Now, we can calculate the work done by multiplying the force with the distance:
Work = Force x Distance
= 7.384 x 10^-2 N x 123,000 m
= 9,093.192 J
So, Lance Armstrong does approximately 9,093.192 joules of work during a 123-km race.
To express this work in nutritional Calories, we need to convert joules into nutritional Calories using the conversion factor given:
1 joule = 2.389 x 10^-4 nutritional Calories
Work in nutritional Calories = Work in joules x Conversion factor
= 9,093.192 J x 2.389 x 10^-4 Cal/J
= 2.177 nutritional Calories
Therefore, the work done by Lance Armstrong during the race is approximately 2.177 nutritional Calories.