Olympiad problems

2009 F = Ma, 20

Tags: 2009 , problem 20

Consider a completely inelastic collision between two lumps of space goo. Lump 1 has mass $m$ and originally moves directly north with a speed $v_\text{0}$. Lump 2 has mass $3m$ and originally moves directly east with speed $v_\text{0}/2$. What is the final speed of the masses after the collision? Ignore gravity, and assume the two lumps stick together after the collision. (A) $7/16 \ v_\text{0}$ (B) $\sqrt{5}/8 \ v_\text{0}$ (C) $\sqrt{13}/8 \ v_\text{0}$ (D) $5/8 \ v_\text{0}$ (E) $\sqrt{13/8} \ v_\text{0}$

2011 F = Ma, 20

Tags: 2011 , problem 20

What is the maximum distance between the particle and the origin? (A) $\text{2.00 m}$ (B) $\text{2.50 m}$ (C) $\text{3.50 m}$ (D) $\text{5.00 m}$ (E) $\text{7.00 m}$

2008 F = Ma, 20

Tags: 2008 , problem 20

The Young’s modulus, $E$, of a material measures how stiff it is; the larger the value of $E$, the more stiff the material. Consider a solid, rectangular steel beam which is anchored horizontally to the wall at one end and allowed to deflect under its own weight. The beam has length $L$, vertical thickness $h$, width $w$, mass density $\rho$, and Young’s modulus $E$; the acceleration due to gravity is $g$. What is the distance through which the other end moves? ([i]Hint: you are expected to solve this problem by eliminating implausible answers. All of the choices are dimensionally correct.[/i]) (a) $h \exp\left( \frac{\rho gL}{E} \right)$ (b) $2\frac{\rho gh^2}{E}$ (c) $\sqrt{2Lh}$ (d) $\frac{3}{2}\frac{\rho gL^4}{Eh^2}$ (e) $\sqrt{3}\frac{EL}{\rho gh}$

2010 F = Ma, 20

Tags: 2010 , problem 20

Consider the following graph of position vs. time, which represents the motion of a certain particle in the given potential. [asy] import roundedpath; size(300); picture pic; // Rectangle draw(pic,(0,0)--(20,0)--(20,15)--(0,15)--cycle); label(pic,"0",(0,0),S); label(pic,"2",(4,0),S); label(pic,"4",(8,0),S); label(pic,"6",(12,0),S); label(pic,"8",(16,0),S); label(pic,"10",(20,0),S); label(pic,"-15",(0,2),W); label(pic,"-10",(0,4),W); label(pic,"-5",(0,6),W); label(pic,"0",(0,8),W); label(pic,"5",(0,10),W); label(pic,"10",(0,12),W); label(pic,"15",(0,14),W); label(pic,rotate(90)*"x (m)",(-2,7),W); label(pic,"t (s)",(11,-2),S); // Tick Marks draw(pic,(4,0)--(4,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(12,0)--(12,0.3)); draw(pic,(16,0)--(16,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(4,15)--(4,14.7)); draw(pic,(8,15)--(8,14.7)); draw(pic,(12,15)--(12,14.7)); draw(pic,(16,15)--(16,14.7)); draw(pic,(20,15)--(20,14.7)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(0,12)--(0.3,12)); draw(pic,(0,14)--(0.3,14)); draw(pic,(20,2)--(19.7,2)); draw(pic,(20,4)--(19.7,4)); draw(pic,(20,6)--(19.7,6)); draw(pic,(20,8)--(19.7,8)); draw(pic,(20,10)--(19.7,10)); draw(pic,(20,12)--(19.7,12)); draw(pic,(20,14)--(19.7,14)); // Path add(pic); path A=(0.102, 6.163)-- (0.192, 6.358)-- (0.369, 6.500)-- (0.526, 6.642)-- (0.643, 6.712)-- (0.820, 6.830)-- (0.938, 6.901)-- (1.075, 7.043)-- (1.193, 7.185)-- (1.369, 7.256)-- (1.506, 7.374)-- (1.644, 7.445)-- (1.840, 7.515)-- (1.958, 7.586)-- (2.134, 7.657)-- (2.291, 7.752)-- (2.468, 7.846)-- (2.625, 7.846)-- (2.899, 7.893)-- (3.095, 8.035)-- (3.350, 8.035)-- (3.586, 8.106)-- (3.860, 8.106)-- (4.135, 8.106)-- (4.371, 8.035)-- (4.606, 8.035)-- (4.881, 8.012)-- (5.155, 7.917)-- (5.391, 7.823)-- (5.665, 7.728)-- (5.960, 7.563)-- (6.175, 7.468)-- (6.332, 7.374)-- (6.528, 7.232)-- (6.725, 7.161)-- (6.882, 6.996)-- (7.117, 6.854)-- (7.333, 6.712)-- (7.509, 6.523)-- (7.666, 6.358)-- (7.902, 6.146)-- (8.098, 5.980)-- (8.274, 5.791)-- (8.451, 5.649)-- (8.647, 5.484)-- (8.882, 5.248)-- (9.196, 5.059)-- (9.392, 4.894)-- (9.628, 4.752)-- (9.824, 4.634)-- (10.118, 4.516)-- (10.452, 4.350)-- (10.785, 4.232)-- (11.001, 4.185)-- (11.315, 4.138)-- (11.648, 4.114)-- (12.002, 4.114)-- (12.257, 4.091)-- (12.610, 4.067)-- (12.825, 4.161)-- (13.081, 4.185)-- (13.316, 4.279)-- (13.492, 4.327)-- (13.689, 4.445)-- (13.826, 4.516)-- (14.022, 4.587)-- (14.159, 4.705)-- (14.316, 4.823)-- (14.532, 4.964)-- (14.669, 5.059)-- (14.866, 5.177)-- (15.062, 5.248)-- (15.278, 5.461)-- (15.474, 5.697)-- (15.650, 5.838)-- (15.847, 6.004)-- (16.043, 6.169)-- (16.258, 6.334)-- (16.415, 6.523)-- (16.592, 6.736)-- (16.788, 6.830)-- (17.063, 7.067)-- (17.357, 7.232)-- (17.573, 7.397)-- (17.808, 7.515)-- (18.063, 7.634)-- (18.358, 7.704)-- (18.573, 7.870)-- (18.887, 7.941)-- (19.142, 8.012)-- (19.358, 8.035)-- (19.574, 8.082)-- (19.770, 8.130); draw(shift(1.8*up)*roundedpath(A,0.09),linewidth(1.5)); [/asy] What is the total energy of the particle? (A) $\text{-5 J}$ (B) $\text{0 J}$ (C) $\text{5 J}$ (D) $\text{10 J}$ (E) $\text{15 J}$