Tutor profile: Chris D.
Subject: Physics (Newtonian Mechanics)
A ball is thrown at a velocity of 12 m/s at an angle of 32 degrees from the horizontal. What are the ball's horizontal and vertical velocities?
Newtonian physics problems are very commonly solved by visualizing the scenario into a free body diagram, which I cannot draw here. But if I could, what I would draw here would be a triangle with the 12.5 m/s written on the hypotenuse, and an "x" and "y" drawn to depict separating that velocity into the horizontal (x) and vertical (y) velocities. The 32 degree angle belongs between the hypotenuse and the "x" axis, as I've described it above. Since I know two pieces of information, the hypotenuse and the angle, using SOH CAH TOA to recall the trigonometric identities that will give me the formulas needed to solve for my unknown quantities. SOH gives me that the "sin 32 = O/H", and since the "opposite" side is y, that leads to "sin 32 = y/12", which can be rearranged into "y = 12 x sin32" = 6.36 m/s. CAH gives me that the "cos 32 = A/H", and since the "adjacent" side is x, that leads to "cos 32 = x/12", which can be rearranged into "x = 12 x cos 32" = 10.18 m/s.
Subject: Basic Chemistry
Mercury has a density of 13.5 g/mL. Determine the number of moles of mercury in 1.0 L of mercury.
For any conversion problem, the trick is to find out where to begin. Look for the number given that has only one unit. In this problem, there are two numbers given: 13.5 and 1.0. Since the 13.5 is a number that has 2 units (or a "compound" unit), that number can easily be used as a conversion factor, so I probably will NOT start with that one. "1.0 L of mercury" has only one unit, so let's start with that one. The reason we try to start with one unit, is so that we can then have each step multiplying by a conversion factor. A conversion factor is a fraction where both the numerator ("top") and denominator ("bottom") of the fraction are equal to each other, but in different units. This means I am NOT multiplying by a fraction that will change the number (multiplying by "1/2", for example, changes the number to something half as large), but by a fraction that will ONLY change the units, cancelling out the unit that I put in the "bottom", and replacing it with the unit I place on top. For example, let's focus on the first step above. Since I am starting with "L", or liters, of mercury, and the density given is in "mL", it would be useful to change (or convert) Liters into mL. This is easily done, since 1000 mL and 1 L are the same thing. So if I multiplied 1.0 L of mercury by the fraction "1000 mL / 1 L", since the bottom of that fraction is in the same "L" unit as the "1.0 L of mercury" that I started with, when I process the multiplication step, those two "L" units will be divided by each other and cancel, replacing the L with "mL". I say "when we process", because that is the second "trick" of conversion problems: set up the problem, step by step, cancelling out units and replacing them until you end up with the unit you want in your answer being the only unit not cancelled before you perform ANY math. In the example above, if I were to follow through with the first step now, I would multiply 1.0 by 1000, and that would be correct to do, but since I haven't set up my next step yet, I would have to write down my answer somewhere to "save" it, and perhaps round early, since I do not have my final answer yet. This would result in too much rounding or writing, or both. So set up your problem, make sure your units all cancel, THEN grab your calculator and do math only once. So, let's add my next steps. 1.0 L of mercury x 1000 mL/1L x 13.5 g/1 mL x 1 mol/200.6 g You see that the first step is to cancel L, replacing with mL, the second step uses the given density to cancel mL and replace with g, and the third step uses the molar mass from the periodic table to cancel g and end with mol. It is of note that the 13.5 and the 200.6 are both numbers that were given as compound units, but whenever we are given a compound unit, the "denominator" unit is always "our of 1" of them, meaning that "13.5 g/mL" is the same as saying "13.5 g is 1 mL", which is where I got the 1's in the work above. You can see that L, mL, g all cancel when I process the steps above, leaving "mol of mercury" as the only units left uncancelled, which means, since that is what I was tasked to find, I have set up all of my steps correctly. The only thing left to do now is my math, which requires multiplying by everything on the numerator side and dividing by everything on the denominator side. You may multiply all the denominators together (perhaps with parentheses) and divide by the total once, but the way that I ensure that all denominators get divided is to do the following steps in my calculator IN THIS ORDER: "1.0 TIMES 1000 EQUALS, DIVIDED by 1 EQUALS, TIMES 13.5 EQUALS, DIVIDED by 1 EQUALS, TIMES 1 EQUALS, DIVIDED by 200.6 EQUALS. ((You may skip any step that multiplies or divides by one, as that doesn't change the answer mathematically, but I always follow through with them to ensure I do not miss a step)) The answer comes out as 67 moles of mercury. The answer actually came out to 67.2981....., but since the "1.0" given in the problem that I started with has only 2 significant figures, I must round back to the second significant figure in my answer, which is the "7", leaving 67 as my numerical answer.
4x + 24 = 36. Solve for x.
My tutoring for a question like this depends on the student. I would typically ask if they've solved problems like this before. If they haven't, and/or it is apparent that this student needs more assistance or even a refresher on the lesson behind it, I actually work backwards. By that I mean that I have them approach this is a puzzle where x=? is the answer, and the puzzle has been created by someone who is trying to hide the answer by making the equation more difficult. So, let's begin by pretending we know what x is. Let's try x = 2. Since the final equation has a 4x and a "+ something", that means that the puzzle builder multiplied AND added to get to the final puzzle. Since these elements have to be added by the order of PEMDAS (which I would explain if the student doesn't know), multiplication is done BEFORE addition. This means that the puzzle maker multiplied by 4, which must be done to both sides. 4 times x gives 4x, and 4 times 2 gives 8, leaving "x = 2" to become "4x = 8". The puzzle maker then added 24, which, again, needs to be done to both sides. Adding 24 to 8 gives 32, making the new equation "4x + 24 = 32". Since the left side of the equation ("4x + 24") is exactly as it should be, my new equation is done, but does not add up to 36, which means the x=2 I started with is the wrong "answer". But building the equation up (backwards) like this, shows the student how the puzzle (or problem) can be built up using PEMDAS laws that do not change. Once they have this foundation, we are left with two options, guess and check (which is very time consuming), or undoing each step in reverse order, which is why PEMDAS must be done backwards while solving. ((If the student knew all of these foundational topics to begin with, I would start here)) Since you have to follow PEMDAS backwards, the addition is the last step, so must be undone first. Undoing addition is subtraction (the opposite function), so subtract 24 to undo ADDING 24. This must be done to both sides, so subtracting 24 from the left side undoes, or cancels out the addition of 24, leaving only the "4x", while subtracting 24 from 36 leaves 12. The new equation then becomes "4x = 12". Since there is no addition or subtraction in this new equation, we can move on to multiplication, where the x is being multiplied by 4. The opposite function of multiplication is division, so dividing both sides of the equation will "undo", or cancel out the multiplication of 4. Dividing 4x by 4 will leave only x, while dividing 12 by 4 leaves 3. So the new equation becomes "x = 3". When you have x by itself on one side, you have now solved for x, and do not need any more steps.
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