A raindrop (mass 1.00 g) is travelling at a speed of 40m/s when it hits the surface of 100 g of water contained in a drinking glass. What is the change in temperature of the water in the glass if we assume that a) all the kinetic energy of the raindrop is converted to energy that changes the water's temperature, and b) the water in the glass and the raindrop was at the same original temperature?
1.00 g is a lot more than a raindrop of water. Its kinetic energy is
(1/2)MV^2 = (1/2)*(1.0)*(1600) = 800J
= 191 calories
That amount of KE gets converted to heat (Q) and added to the internal energy of 101 g of water.
The temperature change is
deltaT = Q/(M*C)
where C is the specific heat of liquid water, 1.0 Cal/g*C
deltaT = 191/101 = 1.89 C
To find the change in temperature of the water in the glass, we can use the equation:
ΔQ = mcΔT
Where:
ΔQ = change in heat energy
m = mass of the water
c = specific heat capacity of water
ΔT = change in temperature
Let's find the change in heat energy (ΔQ) first.
a) Assuming all the kinetic energy is converted to energy that changes the water's temperature:
The kinetic energy of the raindrop can be calculated using the equation:
KE = 0.5mv^2
Where:
m = mass of the raindrop
v = velocity of the raindrop
Substituting the given values:
m = 1.00 g = 0.001 kg (since 1 g = 0.001 kg)
v = 40 m/s
KE = 0.5 * 0.001 kg * (40 m/s)^2
KE = 0.5 * 0.001 kg * 1600 m^2/s^2
KE = 0.8 J
We assume that all the kinetic energy is converted into heat energy, so ΔQ = 0.8 J.
Now, let's calculate the change in temperature (ΔT).
b) Assuming the water in the glass and the raindrop were at the same original temperature:
We can use the equation ΔQ = mcΔT and rearrange it to solve for ΔT:
ΔT = ΔQ / (mc)
Substituting the known values:
m = 100 g = 0.1 kg (since 1 g = 0.001 kg)
c = specific heat capacity of water = 4.18 J/g°C (approximated as 4.18 J/gK)
ΔQ = 0.8 J
ΔT = 0.8 J / (0.1 kg * 4.18 J/gK)
ΔT ≈ 1.92 K
Therefore, the change in temperature of the water in the glass, assuming all the kinetic energy of the raindrop is converted to energy that changes the water's temperature and the water in the glass and the raindrop was at the same original temperature, is approximately 1.92 Kelvin.
To calculate the change in temperature of the water in the glass, we can use the principle of conservation of energy. We'll assume that the water in the glass and the raindrop were at the same original temperature.
a) First, let's calculate the kinetic energy of the raindrop using the formula:
Kinetic energy (KE) = (1/2) * mass * velocity^2
Given:
Mass of the raindrop (m) = 1.00 g = 0.001 kg
Velocity of the raindrop (v) = 40 m/s
Plugging these values into the formula:
KE = (1/2) * 0.001 kg * (40 m/s)^2
KE = 0.02 J
In this case, we assume that all the kinetic energy of the raindrop is converted to energy that changes the water's temperature. So, the change in temperature can be calculated using the specific heat capacity formula:
Change in temperature (ΔT) = (Energy transferred) / (Mass of water * Specific heat capacity of water)
Given:
Mass of water (m_water) = 100 g = 0.1 kg
Specific heat capacity of water (c_water) = 4.18 J/g°C (approximate value for water)
Plugging these values into the formula:
ΔT = 0.02 J / (0.1 kg * 4.18 J/g°C)
ΔT ≈ 0.048°C
Therefore, if all the kinetic energy of the raindrop is converted to energy that changes the water's temperature, the change in temperature of the water in the glass would be approximately 0.048°C.
b) In this scenario, if the water in the glass and the raindrop were at the same original temperature, there would be no change in temperature since there would be no net transfer of thermal energy.