A 4.2 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 1.5 kg of water is raised 0.10°C?
You are not going to die. You just might not get the right answer.
To find 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 to the kinetic energy of the falling rock and then transferred to the water, resulting in an increase in the water's temperature.
Step 1: Calculate the change in potential energy of the rock.
The potential energy of an object near the surface of the Earth is given by the formula:
Potential Energy (PE) = mass (m) * gravitational acceleration (g) * height (h)
In this case, the mass of the rock is 4.2 kg, and the gravitational acceleration is approximately 9.8 m/s^2. Since the rock falls down, the change in height is the height of the cliff.
Step 2: Calculate the kinetic energy of the falling rock.
The kinetic energy of an object is given by the formula:
Kinetic Energy (KE) = (1/2) * mass (m) * velocity^2
Since the rock starts from rest, the initial velocity is zero. Therefore, the initial kinetic energy of the rock is also zero.
Step 3: Calculate the change in kinetic energy of the water.
The change in kinetic energy of the water is equal to the kinetic energy transferred from the rock to the water. Assuming all the rock's kinetic energy is transferred, we can set up the following equation:
Change in Kinetic Energy (KE) = mass of water (m) * specific heat capacity (c) * change in temperature (∆T)
In this case, the mass of the water is 1.5 kg, the specific heat capacity of water is approximately 4186 J/kg°C, and the change in temperature is 0.10°C.
Step 4: Set up the equation for conservation of energy.
The change in potential energy of the rock must be equal to the change in kinetic energy of the water:
Change in PE = Change in KE
Using the formulas from steps 1, 2, and 3, we can write the equation as:
m * g * h = m * c * ∆T
Canceling out the mass (m) from both sides of the equation, we get:
g * h = c * ∆T
Now, we can substitute the known values and solve for the height of the cliff:
(9.8 m/s^2) * h = (4186 J/kg°C) * (0.10°C)
h = (4186 J/kg°C * 0.10°C) / 9.8 m/s^2
h ≈ 42.98 meters
Therefore, the height of the cliff is approximately 42.98 meters.
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 rock as it falls, and then ultimately transferred to the water in the form of thermal energy.
Here's how we can approach the problem step by step:
1. Calculate the gravitational potential energy of the rock at the top of the cliff using the formula:
Potential energy = mass * gravitational acceleration * height
Let's denote the height of the cliff as 'h', and the gravitational acceleration as 'g' (usually taken as 9.8 m/s²).
The potential energy of the rock is given by: P.E. = 4.2 kg * 9.8 m/s² * h
2. At the base of the cliff, all the potential energy of the rock is converted into the kinetic energy of the rock. The kinetic energy (K.E.) is given by the equation:
Kinetic energy = (1/2) * mass * velocity²
Since the rock falls freely under the influence of gravity, its velocity just before hitting the water can be calculated using the equation:
velocity = √(2 * gravitational acceleration * height)
Substituting this velocity value into the kinetic energy equation, we get:
Kinetic energy = (1/2) * 4.2 kg * [(√(2 * 9.8 m/s² * h)]²
3. Now that we know the total kinetic energy of the rock after falling, we can calculate the thermal energy received by the water. The equation relating thermal energy (Q), mass (m), and change in temperature (ΔT) is given as:
Q = m * specific heat capacity * ΔT
For water, the specific heat capacity is approximately 4.18 J/g°C (or 4180 J/kg°C)
Putting all these values, we have:
Q = 1.5 kg * 4180 J/kg°C * 0.10°C
4. According to the principle of conservation of energy, the kinetic energy (K.E.) of the rock is equal to the thermal energy (Q) received by the water. So, we can set the two equations equal to each other:
(1/2) * 4.2 kg * [(√(2 * 9.8 m/s² * h)]² = 1.5 kg * 4180 J/kg°C * 0.10°C
5. Solve the equation for 'h' to find the height of the cliff.
Simplifying the equation, we get:
(√(2 * 9.8 m/s² * h)]² = 1.5 kg * 4180 J/kg°C * 0.10°C * 2 / 4.2 kg
(√(2 * 9.8 m/s² * h)]² = 627 J
2 * 9.8 m/s² * h = 627 J
h = 627 J / (2 * 9.8 m/s²)
Calculate h to get the final answer.
Please note that the unit of height 'h' will depend on the unit of gravitational acceleration 'g' used in the calculation.