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?
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To solve this problem, we need to use the equations of work and power.
(a) The mechanical work done is given by the equation W = F * d * cos(θ), where F is the force applied, d is the displacement, and θ is the angle between the force and displacement vectors. Since the sprinter is starting from rest, the initial velocity is 0. Therefore, we can use the equation W = ΔKE, where ΔKE is the change in kinetic energy.
The change in kinetic energy ΔKE = (1/2) * m * (v^2 - u^2), where m is the mass of the sprinter, v is the final velocity, and u is the initial velocity.
Substituting the values, m = 79 kg, v = 11.0 m/s, and u = 0, we can calculate the mechanical work done by the sprinter.
W = (1/2) * m * (v^2 - u^2)
W = (1/2) * 79 kg * (11.0 m/s)^2
Calculating this expression will give us the mechanical work done by the sprinter in joules (J).
(b) Average power is given by the equation P = W / t, where P is power, W is work done, and t is time taken.
We already calculated the work done in part (a), so we can substitute this value, along with the time taken, to find the average power.
P = W / t
(c) To answer this part, we need to consider that 1 Calorie (with a capital C) is equal to 1000 calories (with a small c), and 1 Calorie is equivalent to 4184 J.
To find the average rate at which the sprinter is burning Calories, we can find the average power and then convert it to Calories.
The average power has already been calculated in part (b). Let's say it is P_avg.
Calories burned (with a capital C) = (P_avg / 4184) * t, where t is the time taken in seconds.
This will give us the energy expended by the sprinter in Calories.
(d) The 75% of food energy that is not converted to mechanical energy is likely converted into other forms, such as heat. During physical activity, the body generates heat that is lost to the surroundings. So, in this case, the remaining 75% of food energy is likely being dissipated as heat.
To solve this problem, we can use the formulas related to work, power, and energy.
(a) To calculate the mechanical work done by the sprinter, we can use the formula:
Work = Force x Distance
In this case, the sprinter's mass can be used to calculate the force using Newton's second law:
Force = Mass x Acceleration
The acceleration can be calculated using the formula:
Acceleration = (Final velocity - Initial velocity) / Time
Using the given values, we can calculate the acceleration:
Acceleration = (11.0 m/s - 0 m/s) / 4.8 s
Acceleration = 11.0 m/s / 4.8 s
Now that we have the acceleration, we can calculate the force:
Force = Mass x Acceleration
Force = 79 kg x (11.0 m/s / 4.8 s)
Now, we can calculate the work done by the sprinter:
Work = Force x Distance
The distance covered by the sprinter is the average of the initial and final velocities multiplied by the time:
Distance = (Final velocity + Initial velocity) / 2 x Time
Distance = (11.0 m/s + 0 m/s) / 2 x 4.8 s
Now, we can calculate the work done:
Work = Force x Distance
(b) To calculate the average power generated by the sprinter, we can use the formula:
Power = Work / Time
(c) To calculate the average rate at which the sprinter is burning Calories, we need to convert the work done to Calories and then divide by the time:
Calories = Work / (Efficiency x 4.18 J/cal)
Average rate of burning Calories = Calories / Time
(d) The other 75% of the food energy being used is likely being lost as heat. This is because no energy conversion is 100% efficient, and some energy is always lost in the form of heat in real-world systems.
Let's now calculate the values step-by-step:
(a) Calculation of mechanical work done by the sprinter:
Acceleration = 11.0 m/s / 4.8 s
Force = 79 kg x (11.0 m/s / 4.8 s)
Distance = (11.0 m/s + 0 m/s) / 2 x 4.8 s
Work = Force x Distance
(b) Calculation of average power generated by the sprinter:
Power = Work / Time
(c) Calculation of average rate at which the sprinter is burning Calories:
Calories = Work / (Efficiency x 4.18 J/cal)
Average rate of burning Calories = Calories / Time
(d) The other 75% of the food energy being used is likely being lost as heat during the conversion process.
a. a = (V-Vo)/t = (11-0)/4.8=2.29 m/s^2.
d = 0.5a*t^2 = 0.5*2.29*(4.8)^2=26.4 m.
Work = F*d = mg*d = (79*9.8) * 26.4 =
20,439 J.
b. P = F*d/t = 20439/4.8 = 4258 J/s =
4258 W.