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 particle is projected with velocity \(v_0\) horizontally from a height \(H\) over a hill, and strikes the hill perpendicularly.

Find \(2gH/v_0 ^ 2\)

Assume the hill rises linearly from the origin at an angle \(\theta = \dfrac\pi6 = \si{30}^\circ\).

A massive particle is launched from ground level with a velocity of magnitude \(v\) and a launch angle of \(\theta\) with respect to the ground.

Suppose a large (essentially infinite) number of launches take place. Over the many trials, \(\theta\) varies uniformly between \(0\) and \(\frac{\pi}{2}\), and \(v\) varies uniformly between 0 and \(v_\text{max}\).

If there is a uniform downward gravitational acceleration \(g\), the expected average distance of the landing point from the launch point (assuming level ground) can be expressed as \(\dfrac{a}{b} \dfrac{v_\text{max}^{2}}{\pi g}\), where \(a\) and \(b\) are coprime positive integers.

Determine \(a+b\).

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 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.

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