A 4.2 kg rock is initially at rest at the top of
a cliff. Assume that 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? The acceleration of gravity is 9.81 m/s
2
and the specific
heat of water is 4186 J/kg ·
◦
C.
Answer in units of m
To solve this problem, we can use the principle of conservation of energy, which states that the potential energy at the top of the cliff is equal to the sum of the kinetic energy transferred to the water and the change in thermal energy of the water.
The potential energy of the rock at the top of the cliff can be calculated using the formula:
PE = mgh
where m is the mass of the rock (4.2 kg), g is the acceleration due to gravity (9.81 m/s^2), and h is the height of the cliff.
The kinetic energy transferred to the water can be calculated using the formula:
KE = (1/2)mv^2
where m is the mass of the rock (4.2 kg) and v is the final velocity of the rock when it hits the water. Since the rock is initially at rest, the final velocity can be calculated using the kinematic equation:
v^2 = u^2 + 2as
where u is the initial velocity (0 m/s), a is the acceleration due to gravity (9.81 m/s^2), and s is the distance fallen (the height of the cliff).
The change in thermal energy of the water can be calculated using the formula:
ΔQ = mcΔT
where m is the mass of the water (1.5 kg), c is the specific heat of water (4186 J/kg °C), and ΔT is the change in temperature (0.10 °C).
Since the kinetic energy transferred to the water is equal to the change in thermal energy of the water, we can equate KE to ΔQ:
(1/2)mv^2 = mcΔT
Now, let's solve for h:
PE = KE + ΔQ
mgh = (1/2)mv^2 + mcΔT
gh = (1/2)v^2 + cΔT
h = [(1/2)v^2 + cΔT] / g
Substituting the values:
h = [(1/2)(0^2 + 2(9.81)(s)) + (4186)(0.10)] / 9.81
Simplifying the equation:
h = [9.81s + 418.6] / 9.81
h = s + 42.63
Therefore, the height of the cliff is s + 42.63 meters.
To solve this problem, we need to understand the concept of gravitational potential energy and the heat transfer equation.
First, let's calculate the amount of potential energy the rock has when it is at the top of the cliff. The formula for gravitational potential energy is:
Potential Energy = mass x acceleration due to gravity x height
Given:
Mass of the rock (m) = 4.2 kg
Acceleration due to gravity (g) = 9.81 m/s^2
Let's assume the height of the cliff is "h" meters.
Potential Energy of the rock = 4.2 kg x 9.81 m/s^2 x h
Next, let's calculate the amount of heat transferred to the water.
The formula for heat transfer is:
Heat = mass x specific heat x change in temperature
Given:
Mass of the water (m) = 1.5 kg
Specific heat of water (c) = 4186 J/kg·°C
Change in temperature (ΔT) = 0.10°C
Heat transferred to the water = 1.5 kg x 4186 J/kg·°C x 0.10°C
According to the problem, the kinetic energy of the rock is transferred entirely to the water, so the potential energy of the rock is equal to the heat transferred to the water. Therefore, we can equate the two equations:
4.2 kg x 9.81 m/s^2 x h = 1.5 kg x 4186 J/kg·°C x 0.10°C
Now, we can solve for the height (h):
h = [1.5 kg x 4186 J/kg·°C x 0.10°C] / [4.2 kg x 9.81 m/s^2]
Calculating the equation:
h = 6289.8 J / 41.382 J/m
h ≈ 151.68 m
Therefore, the height of the cliff is approximately 151.68 meters.