Classical Mechanics

# Vector Kinematics: Level 2-3 Challenges

You wish to swim across a small river to your friend on the other side. The river is 30 meters wide and you begin to swim straight across it at a speed of 1 m/s. The current in the river is 0.5 m/s and so the river carries you downriver as you swim. How far downriver in meters are you when you get to the other side?

This is a heritage physics thought experiment.

Imagine a vet with a tranquilizer gun in the jungle who has just spotted a diseased monkey grasping a tree branch above him. The vet must capture the monkey to treat it for a disease before releasing it back into the wild.

The vet also knows that the monkey will reflexively drop from the branch immediately after he pulls the trigger of his tranquilizer gun.

Where should the vet aim? Where would you aim?

Remember the monkey will free fall to the ground the instant you pull the trigger...

The previous problem makes it clear that our billiard table is deterministic, in that if you know the position and velocity of a ball at one time you can predict the motion of the ball for all time. In reality of course it is impossible to exactly specify the position and velocity of a ball, there is always some measurement error. For many systems, the measurement error doesn't overly matter as the error does not grow with time. Hence if you are within some tolerance to start, you will remain within that tolerance.

For example, consider our ball again. Now, however, instead of knowing that the ball starts exactly at the origin, we only know that it starts somewhere nearby the origin, i.e. within a circle of radius $\epsilon$ of the origin. However, it still has velocity (0,1) to start. The ball bounces off the wall N times. After 4N seconds, the ball is within a circle of radius L of the origin. What is L?

Ever flown on and airplane and seen the blinking lights on top of a radio tower? It may seem odd to build towers tall enough to cause trouble for airplanes, but there's a reason radio towers are tall. The taller the tower, the larger the range over which people can get the signal as FM radio waves are roughly limited by line of sight to the tower. This is why there can be lots of FM stations on the same frequency - they have a limited geographic region. To get an idea of the size of the region, consider a tower on top of a mountain such that the total height of the tower above sea level is 2 km. How far away can you be from the base of the tower/mountain (X in the picture) in kilometers before you lose line of sight to it? Remember - the earth is curved...

Details and assumptions

• The radius of the earth is 6370 km.

Suppose you're sauntering about Saint-Germain, Paris, from one cafe to the next, staying only as long as it takes you to knock down a few Brilliant problems. You're wrapping up at Brasserie Lipp and want to go to Cafè de Flore, which is across the road.

The only problem is: it's pouring rain!

If you want to make it from one cafe to the next while staying as dry as possible, how should you move?

Details

• $v_r$ and $v_w$ are the speeds at which you can run or walk across the road, respectively, with $v_w < v_r$.
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