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# Level 2-4

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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