A 3.6 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 1.1 kg of water is raised 0.14◦C? The acceler- ation of gravity is 9.81 m/s2 and the specific heat of water is 4186 J/kg ·◦ C.
Well, that's quite the physics problem you've got there! Let me see if I can calculate the height of the cliff for you, using the power of humor.
First, let's calculate the change in energy of the rock as it falls. We know its mass (3.6 kg) and the acceleration due to gravity (9.81 m/s^2), so we can use the equation ΔE = mgh, where ΔE is the change in energy, m is the mass, g is the acceleration due to gravity, and h is the height.
Now, the kinetic energy of the rock is transferred entirely to the water, so we can equate the change in energy of the rock to the change in energy of the water. This is given by ΔE = mcΔT, where m is the mass of the water (1.1 kg), c is the specific heat of water (4186 J/kg°C), and ΔT is the change in temperature (0.14°C).
Using these equations, we can set up the following expression:
mgh = mcΔT
Substituting the values we know:
3.6 kg * 9.81 m/s^2 * h = 1.1 kg * 4186 J/kg°C * 0.14°C
Now, solving for h:
h = (1.1 kg * 4186 J/kg°C * 0.14°C) / (3.6 kg * 9.81 m/s^2)
Calculating this, we get:
h ≈ 0.055 meters
So, the height of the cliff is approximately 0.055 meters. It seems like a pretty tiny cliff to me! I hope my humorous approach helped you tackle this problem.
To find the height of the cliff, we can use the principle of conservation of energy.
First, we need to find the initial potential energy of the rock at the top of the cliff. The potential energy can be calculated using the formula:
Potential energy (PE) = mass (m) * gravity (g) * height (h)
where
m = mass of the rock = 3.6 kg
g = acceleration due to gravity = 9.81 m/s^2
h = height of the cliff (unknown)
PE = 3.6 kg * 9.81 m/s^2 * h
PE = 35.196 h Joules
Next, we need to find the final kinetic energy of the rock when it hits the water. The kinetic energy can be calculated using the formula:
Kinetic energy (KE) = 0.5 * mass (m) * velocity^2
The velocity can be found using the principle of conservation of energy. Assuming no other external forces or resistance, all of the potential energy at the top of the cliff is converted to kinetic energy at the bottom of the cliff.
PE = KE
35.196 h = 0.5 * 3.6 kg * v^2
v^2 = (35.196 h * 2) / 3.6
v^2 = 19.554 h
Next, we need to find the velocity (v) of the rock. We can use the formula:
v = √(2gh)
v = √(2 * 9.81 m/s^2 * h)
v = √(19.62 h)
Now, we can substitute the value of v into the equation for KE:
19.554 h = 0.5 * 3.6 kg * (19.62 h)
19.554 h = 35.196 h
h = 19.554 h / 35.196
h ≈ 0.5544 meters
Using the given information, the height of the cliff is approximately 0.5544 meters.
To find the height of the cliff, we need to 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 moving rock and then transferred to the water, increasing its temperature.
First, let's calculate the initial potential energy of the rock at the top of the cliff.
Potential energy (PE) can be calculated using the formula:
PE = m * g * h
where
m = mass of the rock (3.6 kg)
g = acceleration due to gravity (9.81 m/s^2)
h = height of the cliff (unknown)
PE = 3.6 kg * 9.81 m/s^2 * h
Next, let's calculate the kinetic energy (KE) of the moving rock. The kinetic energy is given by:
KE = (1/2) * m * v^2
The rock is initially at rest, so its initial velocity (v) is 0. When it falls, it reaches its final velocity when it hits the water. The formula to find the final velocity is:
v^2 = u^2 + 2 * a * s
where
u = initial velocity (0 m/s)
a = acceleration due to gravity (-9.81 m/s^2, negative because it opposes the motion)
s = distance fallen (h, the height of the cliff)
Simplifying the above equation gives:
v^2 = 0 + 2 * (-9.81 m/s^2) * h
v^2 = -19.62 h
Now we can substitute this value of v^2 into the kinetic energy equation to find the kinetic energy of the rock:
KE = (1/2) * 3.6 kg * (-19.62 h)
The kinetic energy gained by the water is equal to the initial potential energy of the rock. So we have:
KE = PE
Substituting the kinetic energy equation and potential energy equation:
(1/2) * 3.6 kg * (-19.62 h) = 3.6 kg * 9.81 m/s^2 * h
Simplifying the equation:
(1/2) * (-19.62) = 9.81
-9.81 h = 9.81
h = -1
We end up with a negative value for h, which doesn't make physical sense. Therefore, there seems to be an error in the given problem or in the calculations. Please double-check the data and try again.