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2D Kinematics: Level 2-4 Challenges

A projectile is thrown horizontally from a $$20\text{ m}$$ building. If, at the landing point, the velocity vector of the projectile makes an angle of $$45^\circ$$ below the ground, calculate the projectile's initial speed in $$\text{m/s}$$.

Take $$g$$ (the accelaration due to gravity) as $$10 \text{ m/s}^2$$.

A horizontal plane supports a stationary vertical cylinder of radius $$R$$ and a disc $$A$$ attached to the cylinder by a horizontal thread $$AB$$ of length $$L$$ (see figure above, top view). An initial velocity $$v$$ is imparted to the disc, as shown above. How long will it move along the plane until it strikes against the cylinder?

Details and Assumptions:

• Neglect friction at all contact surfaces.
• Do not consider gravitational forces.
• The radius of the disc as compared to the radius of the cylinder is very small.

The image above shows two circular planar portals: a blue portal, with its axis oriented in the vertical direction, and an orange portal, with its axis oriented in the horizontal direction. The blue portal is situated at $$y=0$$, and its $$x$$-coordinate is irrelevant. The orange portal is situated at $$x=0$$, and its height ($$y$$-position) varies as described later.

The rules of portal physics are the following:

• An object enters one portal and emerges from the other
• Portals do not conserve vector momentum or total energy (kinetic + potential)
• Portals do conserve the scalar speed of objects that enter / exit them
• An object entering one portal at its center on a trajectory perpendicular to its axis, will emerge from the other portal in like manner
• Portals do not impart their own velocity to objects which pass through them
• Aside from their own strange properties, portals do not otherwise alter the physics of nearby objects and environments
• It takes zero time to go through a pair of portals

At time $$t = 0$$, two things happen simultaneously:

• A massive ball drops from its initial resting position at height $$h_B$$ and falls under the influence of a uniform downward gravitational acceleration $$g$$. It falls toward the center of the blue portal.

• The orange portal (position defined by its center) begins to descend from its initial vertical position $$h_P$$ at a constant speed $$v_P$$.

Consider the $$x$$-coordinate of the ball at the instant at which it intersects the $$x$$-axis after emerging from the orange portal. The value of $$h_B$$ which maximizes this quantity can expressed as:

${h_{B_\text{max}} = \frac{a}{b} g \left(\frac{h_P}{v_P}\right)^2}.$

If $$a$$ and $$b$$ are coprime positive integers, determine the value of $$a+b$$.

You are trying to steal some water from a water tower. Your friend is going to climb the tower and drill a hole in the side (the bottom is too tough).

Your job is to place a bucket at the initial point of impact of the water. How far (in meters) from the side of the tower should you place your bucket?

Assumptions and Details:

• The cylinder is completely filled with water.

• The hole is drilled at the very bottom of the cylinder's side.

• The hole is $$1 \text{ cm}$$ in diameter.

• The density of water is $$1000 \text{ kg} \cdot \text{m}^{-3}$$.

• Atmospheric pressure is $$10^5 \text{ Pa}$$.

• The acceleration due to gravity is $$10 \text{ m} \cdot \text{s}^{-2}$$.

• Air resistance can be ignored.

• The water cylinder is not air tight at the top.

An object is thrown from point O. In the above figure, the blue dotted lines indicate how the trajectory of a projectile would have been if the air resistance was neglected. If the air resistance is not neglected, which path describes the new trajectory best?

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