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2D Kinematics: Level 2-4 Challenges

         

A little boy swings back and forth on the playground. At the highest point in his swing his speed is zero, and at his lowest point his speed is greatest.

Where in the trajectory will his acceleration be zero?


Caution: The acceleration is the rate of change of velocity and not the rate of change of speed.

A projectile is launched from an initial position \((x,y) = (-\SI{4}{\meter},\SI{0}{\meter})\). The projectile's initial speed is \(12\text{ m/s}\) and it is launched at a \(\SI{75}{\degree}\) angle with respect to the \(x\)-axis. There is a staircase with 4 steps, as pictured above. The top step begins at \((x,y) = (\SI{0}{\meter},\SI{4}{\meter})\) meters and the bottom step ends at \((x,y) = (\SI{4}{\meter},\SI{0}{\meter})\). Each step is \(\SI{1}{\meter}\) wide and \(\SI{1}{\meter}\) deep. The projectile flies over part of the staircase and lands on one of the steps. What is the \(x\) coordinate of the landing point in meters?

Details and Assumptions

  • The gravitational acceleration is in the \(-y\) direction, with a magnitude of \(\SI{9.8}{\meter\per\second\squared}\).

Four boats are on a large lake arranged so that they form the corners of a square of edge length \(\ell = 1\text{ km}.\)

At time zero, each boat starts moving with velocity \(v_\textrm{boat} = \SI[per-mode=symbol]{10}{\kilo\meter\per\hour}\) such that its bow is always pointed directly toward its counterclockwise neighbor.

How long will the boats take to collide (in minutes)?

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 \(a+b\).

Tom and Jerry both have equal top running speeds and are initially at points \(A\) and \(B,\) respectively, separated by a distance of \(d\).

They both spot each other and immediately start running at their top speeds. Jerry runs on a straight line perpendicular to the line \(AB\) and Tom runs in such a way that its velocity always points towards the current location of Jerry.

Let \(r(t)\) denote the distance between Tom and Jerry at time \(t\).

Find \(\displaystyle \lim_{t \to \infty} r(t)\).

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