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2 answers:
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Answer:
a)
b)
c)
d)
Explanation:
Given is the data of variation of temperature with respect to the distance traveled:
Temperature T as a function of distance d:
...................................(1)
(a)
Total change in temperature from the start till the end of the journey:
..............................(2)
where:
= final temperature
= initial temperature
∵In the start of the journey d = 0 miles & at the end of the journey d = 100 miles.
So, correspondingly we have the eq. (2) & (1) as:
(b)
Now, the average rate of change of the temperature, with respect to distance, from the beginning of the trip to the end of the trip be calculated as:
......................(3)
where:
= change in distance
change in temperature with respect to distance
putting the respective values in eq. (3)
(c)
comparing the given function of the temperature with the general equation of a straight line:
We find that we have the slope of the equation as 1 throughout the journey and therefore the rate of change in temperature with respect to distance remains constant.
(d)
comparing the given function of the temperature with the general equation of a straight line:
We find that we have the slope of the equation as 1 throughout the journey and therefore the rate of change in temperature with respect to distance remains constant.
Part 1) Which metal will cool the fastest?
To answer this question, we should have a look at the formula of the heat flow rate, which says "how fast" a material is able to heat/cool:
where:
is the heat exchanged
is the time interval
k is thermal conductivity of the material
A the surface where the exchange of heat occurs
the variation of temperature
x is the thickness of the material
We see that the heat flow rate
is linearly proportional to k, the thermal conductivity of the material. So, the larger k, the fastest the metal will cool.
If we have a look at the thermal conductivity of each metal, we find:
- Aluminium: 237 W/(mK)
- Copper: 401 W/(mK)
- Gold: 314 W/(mK)
- Platinum: 69 W/(mK)
Therefore, copper is the material with highest heat flow rate, so the metal which cools fastest.
Part 2) Which sample of copper demonstrates the greatest increase in temperature
To solve this part, we can have a look at how the amount of heat exchanged Q is related to the increase in temperature:
where m is the mass and Cs the specific heat of the material. Re-arranging the formula, we get
therefore, we see that the increase in temperature is inversely proportional to the mass m. This means that the block that will show the largest increase in temperature is the block with the smallest mass, so the correct answer is A) 0.5 kg.
To answer this question, we should have a look at the formula of the heat flow rate, which says "how fast" a material is able to heat/cool:
where:
k is thermal conductivity of the material
A the surface where the exchange of heat occurs
x is the thickness of the material
We see that the heat flow rate
If we have a look at the thermal conductivity of each metal, we find:
- Aluminium: 237 W/(mK)
- Copper: 401 W/(mK)
- Gold: 314 W/(mK)
- Platinum: 69 W/(mK)
Therefore, copper is the material with highest heat flow rate, so the metal which cools fastest.
Part 2) Which sample of copper demonstrates the greatest increase in temperature
To solve this part, we can have a look at how the amount of heat exchanged Q is related to the increase in temperature
where m is the mass and Cs the specific heat of the material. Re-arranging the formula, we get
therefore, we see that the increase in temperature is inversely proportional to the mass m. This means that the block that will show the largest increase in temperature is the block with the smallest mass, so the correct answer is A) 0.5 kg.