While camping, you boil water to make spaghetti. Your pot contains 2.5 kg of water initially at

While camping, you boil water to make spaghetti. Your pot...

While camping, you boil water to make spaghetti. Your pot contains $2.5 \mathrm{~kg}$ of water initially at $10^{\circ} \mathrm{C}$. You stoke up the campfire, and as a result the water gains energy at an increasing rate: $P=a+b t$, where $a=1.1 \mathrm{~kW}, b=2.3 \mathrm{~W} / \mathrm{s},$ and $t$ is the time in $\mathrm{s}$. To the nearest minute, how long will it take to bring the water to a boil?

Key Concepts

Energy Balance

Energy balance is the principle of equating the energy delivered to a system with the energy required to change its temperature. In heating problems, this involves calculating the required energy to raise the temperature of a substance from its initial temperature to a desired final temperature by using the substance's mass and its specific heat capacity.

Specific Heat Capacity

The specific heat capacity is a property of a material that quantifies the amount of heat energy required to increase the temperature of a unit mass of the material by one degree Celsius. This concept is crucial in thermal calculations because it directly relates the energy input to the resulting change in temperature.

Integration of Variable Power

When the power input to a system is not constant but varies with time, integration is used to determine the total energy supplied over a time interval. By integrating the time-dependent power function, one obtains the cumulative energy delivered, which can then be set equal to the energy required by the system for a specified temperature change.

Unit Conversion

Unit conversion is the process of converting a measurement from one unit system to another, such as converting seconds to minutes. This ensures that the final answer is expressed in a meaningful and practical unit for the context of the problem.

Transcript

00:01 Okay, so the question was asking us to the nearest minutes, how long will it take to bring the water to boil? well, first we know the power is given as a function that is associated with the time, which is a plus b t.

00:16 So the integral of the power function can be equal to the amount of heat, okay? because it's associated with the time.

00:25 So if we do some integral, we'll get, since the power function is equal to a plus b t, so if we do some integral, it will get at plus b times t square over 2 is equal to q.

00:38 Okay? if we multiply by 2 on each side, we'll get 2at plus bt square is equal to 2q.

00:43 Move 2q to the left side, we'll get bt square plus 280 minus 2q is equal to 0 .0.

00:49 Okay.

00:50 Remember, the amount of heat is equal to mc.

00:53 Delta t.

00:53 Am is the mass of the water, c is the specific heat of water.

00:57 Delta t is the changing temperature.

00:58 If we plug in mcd delta t, we'll get bd squared plus 280 minus 2 mc dot t is equal to 0.

01:06 And we'll get tissue have two solutions, which is negative 2a plus or minus square root 2a to the power square plus 4b times 2mc delta t over 2b.

01:18 Okay, so this is from this formula.

01:21 I'll show you guys.

01:24 It's minus b.

01:26 So if you guys study algebra, you guys know what i mean.

01:49 Why list this equation in red leather? so you can just find the b from the equations and find a and c from the equations.

02:01 Okay.

02:02 So, well, in physics problem, most of the time, there's only one solutions in the question.

02:13 Why is that? because for example, for this question, t, the time cannot be negative.

02:19 So the time must be positive.

02:21 That means there's only one solution who satisfy such condition, which is, t should only be equal to negative 2a plus square root 2a to the power square plus 4b times 2mc delta t over 2b.

02:58 So this is the only positive solution we can get.

03:02 Because if you use negative 2a minus square root of these whole things, you'll be negative.

03:08 Okay.

03:08 And the time cannot be negative.

03:10 Because you never say negative one second or negative two seconds...

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