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Dominick H.
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Physics (Newtonian Mechanics)
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Question:

How do you figure out an object's motion in a straight line if you know the forces that act on it?

Dominick H.

The general method is to use Newton's second law of motion, which can be summarized by $$F=ma$$, where $$F$$ is the total (net) force on the object, $$m$$ is the mass of the object, and $$a$$ is the acceleration. Once you are given the forces, you add them up, making sure to note which direction is positive and which is negative (a force the points in the negative direction counts as negative). Once you have the sum of the forces, the acceleration is just that divided by the mass of the object. To learn more about the object's motion, you will need to anti-differentiate the expression or value for $$a$$ once to get a velocity and twice to get position. Remember to also substitute initial conditions (i.e. the position and velocity at time 0 - or any other time for that matter), and then you will be done.

Calculus
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Question:

Why is $$cos(x)$$ the derivative of $$sin(x)$$?

Dominick H.

The best way to answer this question is to visualize it. First, I recommend drawing out the $$sin(x)$$ curve. Then, at $$x=0, \pi/2, \pi, 3\pi/2$$, and $$2\pi$$, plot a tangent line to the curve. Measure the slopes of these lines, and plot them. What you should notice is that each of the slopes falls on the $$y=cos(x)$$ curve. This should give you an intuitive idea. If you want a more rigorous proof, then the most direct method is to use the limit definition. $$y' =lim_{h\to0} \frac{sin(x+h)-sin(x)}{h}=lim_{h\to0} \frac{sin(x)cos(h)+cos(x)sin(h)-sin(x)}{h}=lim_{h\to0} \frac{sin(x)cos(h)-sin(x)+cos(x)sin(h)}{h}= lim_{h\to0}(sin(x)*\frac{cos(h)-1}{h}+cos(x)*\frac{sin(h)}{h})=sin(x)*lim_{h\to0}\frac{cos(h)-1}{h}+ cos(x)*lim_{h\to0}\frac{sin(h)}{h}$$. Since $$lim_{h\to0}\frac{cos(h)-1}{h}=0$$ and $$lim_{h\to0}\frac{sin(h)}{h}= 1$$, both of which you should memorize, the limit approaches $$cos(x)$$. In previous steps, I expanded using the sine of a sum of two angles formula, and then simply rearranged the terms.

Astronomy
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Question:

Can you explain parallax to me and what it is used for?

Dominick H.

Think of it this way. Put one finger about 6 inches from your face, and close your left eye. Then close your right eye. Notice how your finger tends to move when looked against the background? That's the basic idea behind parallax. It's used in astronomy to figure out the distance to nearby stars by comparing how they move with respect to all the other background stars. By measuring how far it moves between say, January and July, astronomers can use trigonometry to calculate the distance to it. Of course, it only works for nearby stars, because for distance stars, the shift will be too small to measure. Even for nearby stars, the parallax angle is very small, which is why it was never noticed in ancient times. If the parallax were observable with ancient tools, people would have realized that the Earth is not the center of the universe long before Copernicus

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