A 3.0 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 1.0 kg of water is raised 0.10 degree celsius.
I found the height to be 14.22 m, but I'm not sure if this is correct. Please tell me if I got the right answer or if I'm doing something wrong.
Your answer looks good!
I got the same answer.
To determine the height of the cliff, we need to use the concept of gravitational potential energy and equate it to the kinetic energy of the falling rock.
The gravitational potential energy of an object near the surface of the Earth is given by the equation:
Potential Energy = mass x acceleration due to gravity x height
The kinetic energy of an object moving with a certain velocity is given by the equation:
Kinetic Energy = (1/2) x mass x velocity^2
In this scenario, the rock falls and its kinetic energy is transferred entirely to the water. This means that the kinetic energy of the rock falling is equal to the increase in thermal energy of the water.
Given:
Mass of the rock (m) = 3.0 kg
Mass of the water (m′) = 1.0 kg
Change in temperature of the water (ΔT) = 0.10°C
Step 1: Calculate the change in thermal energy of the water (Q):
The change in thermal energy can be calculated using the formula:
Q = m′ x specific heat capacity (c) x ΔT
Since we don't have the specific heat capacity of water, we can assume it to be approximately 4,186 J/kg°C.
Q = 1.0 kg x 4,186 J/kg°C x 0.10 °C
Q = 418.6 J
Step 2: Calculate the kinetic energy of the falling rock:
The kinetic energy of the rock can be calculated using the formula stated above:
Kinetic Energy = (1/2) x mass x velocity^2
Since the rock is initially at rest, its initial velocity is 0 m/s.
Kinetic Energy = (1/2) x 3.0 kg x (velocity)^2
Kinetic Energy = (1/2) x 3.0 kg x (final velocity)^2
However, since the kinetic energy of the rock is equal to the thermal energy gained by the water, we have:
(1/2) x 3.0 kg x (final velocity)^2 = 418.6 J
Simplifying the equation:
(final velocity)^2 = (2 x 418.6 J) / 3.0 kg
(final velocity)^2 = 139.43 m^2/s^2
Step 3: Calculate the final velocity of the rock:
Taking the square root of both sides:
final velocity = sqrt(139.43 m^2/s^2)
final velocity = 11.8 m/s (approx.)
Step 4: Calculate the height of the cliff:
Now we can use the equation for gravitational potential energy to find the height of the cliff.
Potential Energy = mass x acceleration due to gravity x height
The mass of the rock is 3.0 kg, and the acceleration due to gravity is approximately 9.8 m/s^2.
(1/2) x (3.0 kg) x (11.8 m/s)^2 = 3.0 kg x 9.8 m/s^2 x height
Simplifying the equation:
1/2 x 140 m^2/s^2 = 29.4 m x height
height = (1/2 x 140 m^2/s^2) / 29.4 m
height = 2.0 m
Therefore, the height of the cliff is approximately 2.0 meters, not 14.22 meters.
To determine the height of the cliff, we can equate the potential energy of the rock at the top of the cliff to the kinetic energy it gains when falling. Then, we can equate the kinetic energy gained by the water to the energy needed to raise its temperature.
First, let's calculate the potential energy of the rock at the top of the cliff. The potential energy is given by the formula:
PE = mgh
where m is the mass of the rock (3.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the cliff (unknown).
PE = (3.0 kg)(9.8 m/s^2)(h)
The potential energy of the rock is entirely converted to kinetic energy when it falls. The kinetic energy is given by the formula:
KE = (1/2)mv^2
where m is the mass of the rock (3.0 kg) and v is the final velocity of the rock just before it hits the water.
Since the rock falls from rest, its initial velocity (u) is zero. So, we can use the equation of motion:
v^2 = u^2 + 2as
where a is the acceleration due to gravity (9.8 m/s^2) and s is the distance traveled by the rock (h).
Using the equation of motion, we have:
v^2 = 0 + 2(9.8 m/s^2)(h)
v^2 = 19.6h
Substituting this value of v^2 into the formula for kinetic energy:
KE = (1/2)mv^2
KE = (1/2)(3.0 kg)(19.6h)
KE = 29.4h
The kinetic energy gained by the water is equal to the energy needed to raise its temperature. This can be calculated using the specific heat capacity (c) of water, which is about 4.18 J/g°C.
The mass of the water is given as 1.0 kg, and the change in temperature is 0.10°C.
The energy needed to raise the temperature can be calculated using the formula:
Energy = mcΔT
Energy = (1.0 kg)(4.18 J/g°C)(0.10°C)
Energy = 0.418 J
Now equating the kinetic energy gained by the water to the energy needed to raise its temperature:
29.4h = 0.418 J
Solving for h:
h = 0.418 J / 29.4 ≈ 0.01422 m
So, the height of the cliff is approximately 0.01422 meters or 14.22 cm.
Your calculation appears to be correct. The height of the cliff is indeed approximately 14.22 meters. Well done!