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Nick V.
Undergraduate Student at MIT
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Physics (Newtonian Mechanics)
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Question:

If a rock is thrown off the side of a cliff 80m high with an initial speed of 30 m/s in the horizontal direction, what is the rock's speed at the bottom of the cliff (just before impact)?

Nick V.
Answer:

Since the initial speed is entirely horizontal, finding the vertical component of the speed at the bottom has the initial condition of starting at 0 m/s in the vertical direction. Therefore using vf^2=vo^2 + 2gy, where vo is zero, g is 10 m/s^2, and y is 80m, vf is calculated then to be 40m/s in the vertical direction. As for the horizontal component of the rock's velocity, there is no horizontal acceleration of the rock (ignoring air resistance), making the final horizontal speed still 30m/s. To find the resultant speed, the horizontal and vertical speeds must be added vectorally, which can be done with the pythagorean theorem since they form a right triangle. Therefore v^2=30^2 + 40^2, leaving the final speed at 50 m/s.

Physics (Electricity and Magnetism)
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Question:

A Point charge of charge +Q resides in the center of a hollow regular tetrahedron of side length a. Find the outward flux through one of the faces of the tetrahedron.

Nick V.
Answer:

Using Gauss's Law, the total flux from an enclosed charge is equal to q/ε. Since the tetrahedron symmetrically encloses the point charge, we can say that 1/4th of the total flux goes through each face of the shape, since there are 4 faces in a tetrahedron. From this, the flux through one face must be (+Q/ε)*(1/4)= +Q/(4ε) where ε is the permittivity of free space.

Calculus
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Question:

Using the Divergence Theorem, find the flux of the velocity field F=<x,y,z> through the surface f(x,y)=4-x^2-y^2 bounded by the x-y plane.

Nick V.
Answer:

The divergence Theorem states that the flux surface integral of a vector valued function over a closed surface is equivalent to the Volume integral of the divergence of the function over the volume enclosed by the given surface. ∯〖F(x,y,z)dS 〗=∰∇⋅F dV The divergence of this function is the sum of the partial derivatives with respect to each variable, so 1+1+1=3. Evaluating the triple integral where z ranges from 0 to 4-x^2-y^2, and x and y are converted to polar coordinates, r ranges from 0 to 2 and θ ranges from 0 to 2π. Evaluating this triple integral gives ∬3(4-r^2 )rdrdθ. Evaluating the next integral gives 12∫_0^2π▒dθ=24 π.

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