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2D Kinematics

Level 2-4

         

Which of the following statements are true for a basketball that is projected diagonally upwards (air resistance cannot be ignored)?

a. The basketball's kinetic energy is zero at the highest point of its motion.

b. The basketball’s average kinetic energy is smaller when traveling upwards than downwards.

c. The time taken for the basketball to travel upwards is less than the time taken to travel downward.

d. The horizontal distance traveled when the basketball is moving upwards is equal to the horizontal distance traveled when the basketball is moving downwards.

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 on the ground where the water that comes out is going to land. How far (in meters) from the side of the tower should you place your bucket?

Vital Statistics:

  • The part of the tower that contains water is a cylinder raised \(20 \text{ m}\) off the ground.

  • The cylinder is \(5 \text{ m}\) tall with \(2 \text{ m}\) in diameter and 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 ground around the tower is flat.

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

Note: Not all information is useful!

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 \(a+b\).

What should be the height \(H\) in meters so that the ball of mass \(\SI{1}{\kilo\gram}\) reaches the point \(A?\)

Take \(g=\SI[per-mode=symbol]{10}{\meter\per\second\squared}.\)

  • There is no friction in this system.
  • The ball is not rolling it is sliding.

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 \(10 \text{ m/s}^2\).

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