In an attempt to exercise more, Ms. Jang runs up the stairs at a constant speed. She climbs to the top of a building with a height of 24 m.
a) Calculate the work done for her to reach the top if she is 62 kg.
b) If she climbs the building in 5.2 minutes, calculate the power of Ms. Jang.
c) Ms. Jang drops her calculator (150 g) from the top of the building to the ground. USING ENERGY RELATED CALCULATIONS, calculate the final velocity of the calculator, just before it hits the ground.
a) The work done by Ms. Jang can be calculated using the formula:
Work = Force x Distance
The force exerted by Ms. Jang can be calculated using her weight:
Force = mass x acceleration due to gravity
Using the given mass of Ms. Jang as 62 kg, and the acceleration due to gravity as 9.8 m/s²:
Force = 62 kg x 9.8 m/s² = 607.6 N
The distance she climbed is given as 24 m. Therefore, the work done is:
Work = 607.6 N x 24 m = 14582.4 J
b) Power is defined as the rate at which work is done, which can be calculated using the formula:
Power = Work / Time
The work done was calculated as 14582.4 J. The time taken to climb the building is given as 5.2 minutes, which must be converted to seconds:
Time = 5.2 minutes x 60 seconds/minute = 312 seconds
Therefore, the power of Ms. Jang is:
Power = 14582.4 J / 312 s = 46.77 W
c) To calculate the final velocity of the calculator, we can use the principle of conservation of energy. The potential energy of the calculator initially at the top of the building will be converted into kinetic energy just before it hits the ground.
The potential energy (PE) at the top of the building is given by:
PE = mass x gravitational acceleration x height
The mass of the calculator is given as 150 g, which must be converted to kg:
Mass = 150 g = 0.15 kg
The gravitational acceleration is 9.8 m/s², and the height is 24 m. Therefore, the potential energy is:
PE = 0.15 kg x 9.8 m/s² x 24 m = 35.28 J
The potential energy will be converted into kinetic energy just before the calculator hits the ground. The equation for kinetic energy (KE) is:
KE = 0.5 x mass x velocity²
The mass is still 0.15 kg, and the final velocity is unknown. Therefore, we need to solve for velocity. Rearranging the equation gives:
velocity = √(2 x KE / mass)
Substituting the potential energy calculated earlier:
velocity = √(2 x 35.28 J / 0.15 kg)
velocity ≈ √(470.4 m²/s²) ≈ 21.7 m/s
Therefore, the final velocity of the calculator just before it hits the ground is approximately 21.7 m/s.
a) To calculate the work done by Ms. Jang, we need to use the formula:
Work = Force x Distance x cos(theta)
Since she is running up the stairs at a constant speed, there is no vertical acceleration. This means that the net force in the vertical direction is zero.
The weight of Ms. Jang can be calculated using the formula:
Weight = mass x acceleration due to gravity
Weight = 62 kg x 9.8 m/s^2 = 607.6 N
Since the force and displacement are in the same direction (vertical), the angle between them is 0 degrees. Therefore, cos(theta) = 1.
Now we can calculate the work:
Work = 607.6 N x 24 m x 1 = 14582.4 J
Therefore, the work done by Ms. Jang to reach the top of the building is 14582.4 Joules.
b) Power is defined as the rate at which work is done. The formula for calculating power is:
Power = Work / Time
First, we need to convert the time from minutes to seconds:
5.2 minutes = 5.2 x 60 seconds = 312 seconds
Now we can calculate the power:
Power = 14582.4 J / 312 s = 46.76 Watts
Therefore, the power of Ms. Jang is approximately 46.76 Watts.
c) To calculate the final velocity of the calculator just before it hits the ground, we can use the principle of conservation of mechanical energy.
The initial mechanical energy is equal to the potential energy at the top of the building, and the final mechanical energy is equal to the kinetic energy just before hitting the ground.
The potential energy can be calculated using the formula:
Potential Energy = mass x acceleration due to gravity x height
Potential Energy = 0.150 kg x 9.8 m/s^2 x 24 m = 35.28 J
The final kinetic energy is equal to the potential energy at the top, so the final mechanical energy is also 35.28 J.
The kinetic energy can be calculated using the formula:
Kinetic Energy = 0.5 x mass x velocity^2
35.28 J = 0.5 x 0.150 kg x velocity^2
Now we can solve for the velocity:
velocity^2 = (2 x 35.28 J) / 0.150 kg
velocity^2 = 470.4 m^2/s^2
velocity = sqrt(470.4 m^2/s^2) = 21.7 m/s
Therefore, the final velocity of the calculator just before it hits the ground is approximately 21.7 m/s.