A 79-kg sprinter accelerates from rest to a speed of 11.0 m/s in 4.8 s.
(a) Calculate the mechanical work done by the sprinter during this time.
J
(b) Calculate the average power the sprinter must generate.
W
(c) If the sprinter converts food energy to mechanical energy with an efficiency of 25%, at what average rate is he burning Calories?
(d) What happens to the other 75% of the food energy being used?
This answer has not been graded yet.
See previous post.
To answer these questions, we will use the relevant equations of work, power, and efficiency. Let's break down each question step by step:
(a) To calculate the mechanical work done by the sprinter, we can use the formula:
Work = Force x Distance
Since the sprinter is accelerating from rest to a certain speed, we need to use the equation of motion:
v = u + at
where:
v is the final velocity (11.0 m/s),
u is the initial velocity (0 m/s),
a is the acceleration, and
t is the time taken (4.8 s).
Rearranging the equation to solve for acceleration gives us:
a = (v - u) / t
Substituting the given values:
a = (11.0 m/s - 0 m/s) / 4.8 s = 2.29 m/s^2
Now, we can calculate the force exerted by the sprinter using Newton's second law:
Force = Mass x Acceleration
Substituting the given mass:
Force = 79 kg x 2.29 m/s^2 = 182.91 N
Finally, we can calculate the work done using the equation of work:
Work = Force x Distance
Since the sprinter starts from rest, the distance traveled can be found using the equation for linear displacement:
Distance = (1/2) x acceleration x time^2
Substituting the given values:
Distance = (1/2) x 2.29 m/s^2 x (4.8 s)^2 = 27.84 m
Substituting the values, we get:
Work = 182.91 N x 27.84 m ≈ 5086 J
Therefore, the mechanical work done by the sprinter during this time is approximately 5086 Joules (J).
(b) To calculate the average power the sprinter must generate, we can use the formula:
Power = Work / Time
Substituting the known values:
Power = 5086 J / 4.8 s ≈ 1059 W
Therefore, the average power the sprinter must generate is approximately 1059 Watts (W).
(c) If the sprinter converts food energy to mechanical energy with an efficiency of 25%, we can calculate the average rate at which he is burning calories.
First, we need to find the total energy burned by the sprinter, which is equal to the mechanical work done. We already calculated this in part (a) as 5086 J.
Since 1 calorie is equal to 4.184 J, we can convert the energy burned from joules to calories:
Energy burned (in calories) = Energy burned (in joules) / 4.184
Substituting the known value:
Energy burned (in calories) = 5086 J / 4.184 ≈ 1215 calories
Finally, we need to divide the calories burned by the time taken to find the average rate at which he is burning calories:
Average rate of burning calories = Energy burned (in calories) / Time
Substituting the known value:
Average rate of burning calories = 1215 calories / 4.8 s ≈ 253.13 cal/s
Therefore, the average rate at which the sprinter is burning calories is approximately 253.13 calories per second (cal/s).
(d) The remaining 75% of the food energy that is not converted to mechanical energy by the sprinter is released as heat. This heat is usually dissipated from the body through various means, such as sweating or increased body temperature. The efficiency of the conversion process determines how much energy is converted into useful mechanical work, with the rest being lost as heat. In this case, 25% of the food energy is converted into useful mechanical work, while the remaining 75% is released as heat.
To solve this problem, we can use the formulas for work, power, and energy.
(a) The work done by the sprinter can be calculated using the formula:
Work = force × distance
However, since we are not given the force or distance directly, we need to find them using the known information. We can use the equation of motion:
v = u + at
where v is the final velocity, u is the initial velocity (0 m/s in this case), a is the acceleration, and t is the time.
Rearranging this equation to solve for acceleration:
a = (v - u) / t
Substituting the given values:
a = (11.0 m/s - 0 m/s) / 4.8 s = 2.29 m/s^2
Now, we can calculate the force using Newton's second law:
Force = mass × acceleration
Force = 79 kg × 2.29 m/s^2 = 182.91 N
The sprinter's work can now be calculated:
Work = Force × distance
Since the sprinter starts from rest, the initial velocity is 0 m/s. Using the formula for average velocity:
Average velocity = (0 + 11.0) / 2 = 5.5 m/s
Let's assume the sprinter covers a distance of x meters. The time taken to reach the final velocity of 11.0 m/s is 4.8 seconds.
Using the equation of motion to calculate the distance covered:
x = ut + (1/2)at^2
x = 0 + (1/2)(2.29 m/s^2)(4.8 s)^2 = 27.63 m
Work = 182.91 N × 27.63 m = 5051.41 J
The mechanical work done by the sprinter is approximately 5051.41 J.
(b) The average power can be calculated using the formula:
Power = Work / Time
Power = 5051.41 J / 4.8 s = 1052.38 W
The average power generated by the sprinter is approximately 1052.38 W.
(c) To calculate the rate at which the sprinter is burning Calories, we need to calculate the energy consumption. The energy consumption can be determined using the formula:
Energy consumption = Work / efficiency
Given efficiency = 25%, we can convert it to decimal form as 0.25.
Energy consumption = 5051.41 J / 0.25 = 20205.64 J
To convert energy from J to Calories, we can use the conversion factor: 1 Calorie = 4.184 J.
Energy consumption = 20205.64 J / 4.184 J/Calorie = 4830.45 Calories
Therefore, the sprinter is burning calories at an average rate of approximately 4830.45 Calories.
(d) The other 75% of the food energy being used is lost as waste heat. This is due to the inefficiency of the conversion process from food energy to mechanical energy. The body cannot convert all the energy from food into useful mechanical work, and some of it is lost as heat.