Project
Question: 01
The potential energy (PE) of an object at a certain height is given by:
PE = mgh
where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object.
The kinetic energy (KE) of an object is given by:
KE = (1/2)mv^2
where m is the mass of the object, and v is its velocity.
At point U, the ball has only potential energy, which is converted into kinetic energy at point V. According to the problem, the kinetic energy of the ball at point V is three times its potential energy at point U.
So, we can write:
KE at point V = 3 * PE at point U (1/2)mv^2 = 3mgh
We can cancel out the mass (m) from both sides of the equation:
(1/2)v^2 = 3gh
Solving for h, we get:
h = (1/2) * (v^2/3g)
We are given that the mass of the ball is 1.2 kg. We need to find the height (h) of point V above the surface of the earth. We can assume that the velocity of the ball at point V is equal to the velocity it would have gained by falling freely from point U to point V.
To find the velocity of the ball at point V, we can use the conservation of energy principle:
PE at point U = KE at point V
mgh = (1/2)mv^2
Simplifying, we get:
v = sqrt(2gh)
Substituting this value of v in the expression for h, we get:
h = (1/2) * [(sqrt(2gh))^2/3g] h = (1/2) * (2h/3) h = h/3
Solving for h, we get:
h = 3 * mgh/2mv^2 h = 3 * 1.2 kg * 9.81 ms^-2 * h / [2 * (sqrt(2 * 9.81 ms^-2 * h))]^2
Simplifying and solving for h, we get:
h = 61.08 meters
Therefore, the height of point V above the surface of the earth is approximately 61.08 meters.
Question: 02
We can use the formula for calculating heat transfer through a composite wall to solve this problem:
Q = U * A * deltaT
where Q is the heat transfer rate, U is the overall heat transfer coefficient, A is the area of the wall, and deltaT is the temperature difference across the wall.
To find U, we need to calculate the individual thermal resistances of the three layers of the wall:
R_wood = thickness_wood / (Kw * area) R_cement = thickness_cement / (Kc * area) R_brick = thickness_brick / (Kb * area)
where thickness_wood, thickness_cement, and thickness_brick are the thicknesses of the respective layers.
The overall thermal resistance is the sum of the individual resistances:
R_total = R_wood + R_cement + R_brick
The overall heat transfer coefficient is the reciprocal of the total thermal resistance:
U = 1 / R_total
Now we can plug in the values:
thickness_wood = 0.025 m thickness_cement = 0.02 m thickness_brick = 0.25 m Kw = 0.125 Wm-1°C-1 Kc = 1.5 Wm-1°C-1 Kb = 1 Wm-1°C-1 area = 140 m2 deltaT = 22°C - (-10°C) = 32°C
R_wood = 0.025 / (0.125 * 140) = 0.001786 R_cement = 0.02 / (1.5 * 140) = 0.0000952 R_brick = 0.25 / (1 * 140) = 0.00179
R_total = R_wood + R_cement + R_brick = 0.00367
U = 1 / R_total = 272.14 Wm-2°C-1
Finally, we can calculate the heat transfer rate:
Q = U * A * deltaT = 272.14 * 140 * 32 = 1,207,904.64 W
Therefore, the rate at which the energy is extracted from the ground surface on average is 1,207,904.64 W.