Dumping sealed containers from ship decks that are 2 meters above the water surface. With a final depth of 100 meters.
Find the final velocity of the container on impact.
1. Container in the shape of right circular cylinders, containers of .5 meters diameter and 1.0 meter height.
2. The density of the waste material is 1,200kg/m^3
To find the final velocity of the container on impact, we can use the principle of conservation of energy, assuming no external forces apart from gravity are acting on the container.
First, let's calculate the potential energy of the container at the initial and final positions:
1. Initial position: The potential energy at the initial position is given by the formula P.E. = m * g * h, where m is the mass, g is the acceleration due to gravity, and h is the initial height above the water surface. Since the container is sealed, we need to consider the density of the material to calculate the mass. The volume of a right circular cylinder can be calculated using the formula V = π * r^2 * h, where r is the radius and h is the height. In our case, the radius is half of the diameter, so r = 0.5 * 0.5 = 0.25 meters. Therefore, the volume of the container is V = π * 0.25^2 * 1 = 0.1963 m^3. The mass of the container can be calculated by multiplying the volume by the density: m = V * density = 0.1963 m^3 * 1200 kg/m^3 = 235.56 kg. Now, we can calculate the potential energy at the initial position: P.E. = m * g * h = 235.56 kg * 9.8 m/s^2 * 2 m = 4611.4 Joules.
2. Final position: The potential energy at the final position can be calculated in the same way. The height above the water surface is now 100 meters: P.E. = m * g * h = 235.56 kg * 9.8 m/s^2 * 100 m = 230508 Joules.
Since energy is conserved, the difference between the initial potential energy and the final potential energy is equal to the kinetic energy gained by the container:
Kinetic energy gained = P.E. (initial) - P.E. (final) = 4611.4 Joules - 230508 Joules = -226896.6 Joules (negative because the potential energy decreases)
Now, let's calculate the velocity of the container using the kinetic energy formula:
Kinetic energy = 0.5 * m * v^2, where v is the final velocity of the container.
-226896.6 Joules = 0.5 * 235.56 kg * v^2
Solving for v^2, we get:
v^2 = (-226896.6 Joules) * 2 / 235.56 kg = -3845.64 m^2/s^2
Since the velocity cannot be negative, we take the positive square root:
v = sqrt(3845.64 m^2/s^2) = 61.98 m/s
Therefore, the final velocity of the container on impact with a depth of 100 meters is approximately 61.98 m/s.