Olympiad problems

2008 F = Ma, 6

Tags: 2008 , Problem 6

A cannon fires projectiles on a flat range at a fixed speed but with variable angle. The maximum range of the cannon is $L$. What is the range of the cannon when it fires at an angle $\frac{\pi}{6}$ above the horizontal? Ignore air resistance. (a) $\frac{\sqrt{3}}{2}L$ (b) $\frac{1}{\sqrt{2}}L$ (c) $\frac{1}{\sqrt{3}}L$ (d) $\frac{1}{2}L$ (e) $\frac{1}{3}L$

2009 F = Ma, 6

Tags: 2009 , Problem 6

An object is thrown with a fixed initial speed $v_\text{0}$ at various angles $\alpha$ relative to the horizon. At some constant height $h$ above the launch point the speed $v$ of the object is measured as a function of the initial angle $\alpha$. Which of the following best describes the dependence of $v$ on $\alpha$? (Assume that the height h is achieved, and assume that there is no air resistance.) (A) $v$ will increase monotonically with $\alpha$. (B) $v$ will increase to some critical value $v_{max}$ and then decrease. (C) $v$ will remain constant, independent of $\alpha$. (D) $v$ will decrease to some critical value $v_{min}$ and then increase. (E) None of the above.

2010 F = Ma, 6

Tags: 2010 , Problem 6

A projectile is launched across flat ground at an angle $\theta$ to the horizontal and travels in the absence of air resistance. It rises to a maximum height $H$ and lands a horizontal distance $R$ away. What is the ratio $H/R$? (A) $\tan \theta$ (B) $2 \tan \theta$ (C) $\frac{2}{\tan \theta}$ (D) $\frac{1}{2}\tan \theta$ (E) $\frac{1}{4}\tan \theta$

2011 F = Ma, 6

Tags: 2011 , Problem 6

A child is sliding out of control with velocity $v_\text{c}$ across a frozen lake. He runs head-on into another child, initially at rest, with $3$ times the mass of the first child, who holds on so that the two now slide together. What is the velocity of the couple after the collision? (A) $2v_\text{c}$ (B) $v_\text{c}$ (C) $v_\text{c}/2$ (D) $v_\text{c}/3$ (E) $v_\text{c}/4$