A 3.5 kg rock is initially at rest at the top of a cliff. Assuming the rock falls into the sea at the foot of the cliff and that its kinetic energy is transferred entirely to the water, how high is the cliff if the temperature of 0.88 kg of water is raised 0.10°C? (Neglect the heat capacity of the rock.)
To determine the height of the cliff, we can use the principle of conservation of energy. We know that the initial potential energy of the rock at the top of the cliff is equal to the final kinetic energy of the rock in the water.
The potential energy (PE) of an object at a certain height h is given by the formula:
PE = m * g * h
Where m is the mass of the object, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
In this case, the initial potential energy is equal to the final kinetic energy of the rock, because all the kinetic energy gets transferred to the water. So, we can equate the two energies:
PE = KE
m * g * h = (1/2) * m * v²
Where v is the velocity of the rock just before hitting the water. We don't have the value for v, but we can calculate it using the concept of conservation of energy.
The total energy of the system (rock + water) is conserved. The initial energy of the system is the potential energy of the rock at the top of the cliff, and the final energy is the sum of the final kinetic energy of the rock and the thermal energy absorbed by the water.
Initial energy = Final energy
m * g * h = (1/2) * m * v² + Q
Where Q is the thermal energy absorbed by the water. We can calculate Q using the formula:
Q = mCΔT
Where m is the mass of the water, C is the specific heat capacity of water (approximately 4186 J/kg°C), and ΔT is the change in temperature. Substituting this into the equation:
m * g * h = (1/2) * m * v² + m * C * ΔT
Now, we can substitute the given values into the equation. The mass of the rock (m) is 3.5 kg, the mass of the water (m) is 0.88 kg, the value of g is 9.8 m/s², the specific heat capacity of water (C) is 4186 J/kg°C, and the change in temperature (ΔT) is 0.10°C.
Plugging in the values:
(3.5 kg) * (9.8 m/s²) * h = (1/2) * (3.5 kg) * v² + (0.88 kg) * (4186 J/kg°C) * (0.10°C)
Simplifying the equation:
34.3 h = (1.75 v²) + 3651.52
Since we don't have the value of v, we need to find a way to eliminate it from the equation. We can use the fact that the kinetic energy of the rock is transferred entirely to the water. The kinetic energy (KE) of an object is given by the formula:
KE = (1/2) * m * v²
Substituting this into the equation:
34.3 h = 2 * (KE) + 3651.52
34.3 h = 2 * (1/2) * (3.5 kg) * v² + 3651.52
34.3 h = 3.5 v² + 3651.52
Now we have two equations:
34.3 h = (1.75 v²) + 3651.52
34.3 h = 3.5 v² + 3651.52
These two equations can be solved simultaneously to find the value of h, which represents the height of the cliff.
To determine the height of the cliff, we can use the principle of conservation of energy. The potential energy of the rock at the top of the cliff is converted into the kinetic energy of the falling rock and then transferred to the water, raising its temperature.
The potential energy of the rock at the top of the cliff is given by the equation:
PE = m * g * h
Where:
PE is the potential energy
m is the mass of the rock (3.5 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height of the cliff (unknown)
The kinetic energy of the rock when it reaches the water is given by the equation:
KE = (1/2) * m * v^2
Where:
KE is the kinetic energy
m is the mass of the rock (3.5 kg)
v is the velocity of the rock (unknown)
Since the kinetic energy of the rock is transferred entirely to the water, we can equate the potential energy and kinetic energy equations:
m * g * h = (1/2) * m * v^2
Simplifying and solving for the velocity (v), we get:
v = sqrt(2 * g * h)
We can now substitute the value of v into the equation for the kinetic energy (KE):
KE = (1/2) * m * (sqrt(2 * g * h))^2
KE = m * g * h
Since the kinetic energy transferred to the water is used to raise its temperature, we can equate the kinetic energy to the heat energy:
KE = m_water * c * ΔT
Where:
m_water is the mass of the water (0.88 kg)
c is the specific heat capacity of water (4.184 J/g°C or 4184 J/kg°C)
ΔT is the change in temperature (0.10°C or 0.10 K)
Substituting the values, we get:
m * g * h = m_water * c * ΔT
Simplifying, we can solve for the height (h):
h = (m_water * c * ΔT) / (m * g)
Plugging in the given values:
h = (0.88 kg * 4184 J/kg°C * 0.10 K) / (3.5 kg * 9.8 m/s^2)
Calculating gives:
h ≈ 0.25 meters
Therefore, the height of the cliff is approximately 0.25 meters.