Classical Mechanics
# Vector Kinematics

An NBA basketball player want to score 3 points, so he shoots the ball right before the 3-point-line (which is \(d=6.25~\mbox{m}\) horizontally away from the basket). The difference in height from where he releases the ball and the basket is roughly \(h=1~\mbox{m}\). Given the gravitational acceleration \(a_g=-9.8~\mbox{m/s}^2\), at what angle to the horizontal **in degrees** did he shoot the ball if the horizontal speed of the ball is \(8.71~\mbox{m/s}\)?

**Details and assumptions**

- You may neglect air resistance.

*Flappy Bird*, the player's objective is to get a bird through some pipes. Sounds simple, right?

Wrong. The bird is a odd one that only flaps it's wings when you touch your screen. This causes a small burst in speed sending it up, and then causing to plummet, requiring you too tap the screen again. This incessant screen-tapping is especially difficult when you get to the pipes, because maintaining a level height is feat which requires quite a lot of finesse.

Now, every time you tap the screen, the bird is launched up, and comes back down to its original height after just \(1\) second. If the game obeys the laws of physics, and takes place on Earth, (\(g=9.8 \text{ m/s}^2\)), and the small bird has a mass of about \(500\) grams, find the momentum of the bird right after you touch the screen.

This problem is part of the Science of Apps! series

Christopher accidentally fell inside a cylindrical well of depth 15 meters and radius 8 meters. Jumping from the center of the bottom of the well, what is the minimum launch speed that Christopher needs to achieve, in order to jump out of the well?

Consider Christopher as a particle.

Take \( g = 10 m / s ^ 2 \)

Four beetles, call them John, Paul, George, and Ringo, start off at the corners of a square room 2 meters on a side. Looking down into the room from above, John is in the upper left, Paul in the upper right, George in the lower right, and Ringo in the lower left. The beetles start crawling towards the next beetle, i.e. John crawls directly at Paul, Paul crawls directly at George, George crawls directly at Ringo, and Ringo crawls directly at John. Each beetle's speed is \(0.01~\mbox{m/s}\).

How far has one beetle traveled **in meters** when all the beetles meet in the middle?

You have a potato cannon, located 30 m from the start of a hill. The hill is inclined at \(\theta = \pi/6\) radians, and your cannon fires potatoes with \(v_0 = 40\) m/s. If you want to hit a target located 80 m up the hill (measured along the hill), what is the smallest angle, \(\phi_\text{min}\), to which you can set your cannon (in radians)?

**Details**

- \(g = 9.81\) m/s

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