A person of mass 66 kilograms walked up an inclined plane to a height of 450 meters in five minutes. Determine their average power throughout the walk.
Let’s begin by sketching a diagram to model the scenario. We are told that a person walks up an inclined plane to a height of 450 meters in five minutes. We are told that they have a mass of 66 kilograms and need to calculate the average power throughout the walk. We know that we can calculate the power 𝑝 by multiplying a force 𝐹 by a velocity 𝑣. We will begin by calculating the average velocity 𝑣 by dividing the total displacement by the total time.
As there are 60 seconds in a minute and five multiplied by 60 is 300, the total time is 300 seconds. We can calculate the total displacement 𝑑 using our knowledge of right angle trigonometry. We will begin by letting the angle of inclination be 𝛼. The sine ratio tells us that the sine of angle 𝛼 is equal to the opposite over the hypotenuse. In this question, sin 𝛼 is equal to 450 over 𝑑. And this means that the total displacement 𝑑 is equal to 450 over sin 𝛼. The average velocity is therefore equal to 450 over sin 𝛼 divided by 300. Dividing the numerator and denominator by 150, we have three over sin 𝛼 divided by two. And this simplifies to three over two sin 𝛼. The average power throughout the walk is therefore equal to some force 𝐹 multiplied by three over two sin 𝛼.
We will now use Newton’s second law, 𝐹 equals 𝑚𝑎, to determine the value of the force 𝐹. We know that at any point on their journey, the person will exert a downward force equal to their weight. This is equal to their mass of 66 kilograms multiplied by the acceleration due to gravity 𝑔. Once again, using our knowledge of right angle trigonometry, we see that the force acting down the plane will be equal to 66𝑔 multiplied by sin 𝛼.
We are trying to calculate the force 𝐹 acting up the plane, and assuming that the person is moving at a constant velocity, we know that the acceleration 𝑎 is equal to zero meters per second squared. Taking the positive direction to be the direction of travel, the sum of our forces is 𝐹 minus 66𝑔 multiplied by sin 𝛼. This is equal to a mass of 66 kilograms multiplied by an acceleration of zero. The right-hand side of our equation is equal to zero. And adding 66𝑔 multiplied by sin 𝛼 to both sides, we have 𝐹 is equal to 66𝑔 multiplied by sin 𝛼.
The average power 𝑝 is therefore equal to 66𝑔 sin 𝛼 multiplied by three over two sin 𝛼. We can divide the numerator and denominator by sin 𝛼 and also by two, giving us 33𝑔 multiplied by three. This is equal to 99𝑔. And taking 𝑔 to be 9.8 meters per second squared, we have a power 𝑝 equal to 970.2 watts. Whilst we could leave our answer in these units, we could also give it in horsepower, recalling that one horsepower is equal to 735 watts. Dividing 970.2 by 735 gives us 1.32 horsepower. This is the average power throughout the walk.