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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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