Found problems: 26
2008 F = Ma, 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) $\frac{1}{3}L$
2008 F = Ma, 23
Consider two uniform spherical planets of equal density but unequal radius. Which of the following quantities is the same for both planets?
(a) The escape velocity from the planet’s surface.
(b) The acceleration due to gravity at the planet’s surface.
(c) The orbital period of a satellite in a circular orbit just above the planet’s surface.
(d) The orbital period of a satellite in a circular orbit at a given distance from the planet’s center.
(e) None of the above.
2008 F = Ma, 4
What is the maximum displacement from start for the toy car?
(a) $\text{3 m}$
(b) $\text{5 m}$
(c) $\text{6.5 m}$
(d) $\text{7 m}$
(e) $\text{7.5 m}$
2008 F = Ma, 5
Which of the following acceleration [i]vs.[/i] time graphs most closely represents the acceleration of the toy car?
[asy]
size(300);
Label f;
f.p=fontsize(8);
xaxis(0,3);
yaxis(-2,2);
label("Time (s)",(2.8,0),0.5*N);
label(rotate(90)*"Acceleration",(-0.2,0),W);
label("$0$",(-0,0),SW,fontsize(9));
label("1",(1,0),2*S,fontsize(9));
label("2",(2,0),2*S,fontsize(9));
label("3",(3,0),2*S,fontsize(9));
draw((0.5,0)--(0.5,-0.1));
draw((1,0)--(1,-0.1));
draw((1.5,0)--(1.5,-0.1));
draw((2,0)--(2,-0.1));
draw((2.5,0)--(2.5,-0.1));
draw((3,0)--(3,-0.1));
label("(a)",(1.5,-2),N);
pair A, B, C, D, E, F;
A = (0,1);
B = (1,1);
C = (1,0);
D = (1.5,0);
E = (1.5, 0.5);
F = (3, 0.5);
draw(A--B--C--D--E--F);
real x=6;
Label f;
f.p=fontsize(8);
draw((x+3,0)--(x+0,0));
draw((x,-2)--(x,2));
label("Time (s)",(x+2.8,0.03),0.5*N);
label(rotate(90)*"Acceleration",(x-0.2,0),W);
label("$0$",(x+0,0),SW,fontsize(9));
label("1",(x+1,0),2*S,fontsize(9));
label("2",(x+2,0),2*S,fontsize(9));
label("3",(x+3,0),2*S,fontsize(9));
draw((x+0.5,0)--(x+0.5,-0.1));
draw((x+1,0)--(x+1,-0.1));
draw((x+1.5,0)--(x+1.5,-0.1));
draw((x+2,0)--(x+2,-0.1));
draw((x+2.5,0)--(x+2.5,-0.1));
draw((x+3,0)--(x+3,-0.1));
label("(b)",(x+1.5,-2),N);
/*The lines*/
pair G, H, I, J, K, L;
G = (x+0,1);
H = (x+1,1);
I = (x+1,0);
J = (x+1.5,0);
K = (x+1.5, -0.5);
L = (x+3, -0.5);
draw(G--H--I--J--K--L);[/asy][asy]
size(300);
Label f;
f.p=fontsize(8);
xaxis(0,3);
yaxis(-2,2);
label("Time (s)",(2.8,0),0.5*N);
label(rotate(90)*"Acceleration",(-0.1,0),W);
label("$0$",(-0,0),SW,fontsize(9));
label("1",(1,0),2*S,fontsize(9));
label("2",(2,0),2*S,fontsize(9));
label("3",(3,0),2*S,fontsize(9));
draw((0.5,0)--(0.5,-0.1));
draw((1,0)--(1,-0.1));
draw((1.5,0)--(1.5,-0.1));
draw((2,0)--(2,-0.1));
draw((2.5,0)--(2.5,-0.1));
draw((3,0)--(3,-0.1));
label("(c)",(1.5,-2),N);
pair A, B, C, D, E, F;
A = (0,0.5);
B = (1,0.5);
C = (1,0);
D = (1.5,0);
E = (1.5, -1);
F = (3, -1);
draw(A--B--C--D--E--F);
real x = 6;
Label f;
f.p=fontsize(8);
draw((x+3,0)--(x+0,0));
draw((x,-2)--(x,2));
label("Time (s)",(x+3.4,0),0.5*N);
label(rotate(90)*"Acceleration",(x-0.2,0),W);
label("$0$",(x+0,0),SW,fontsize(9));
label("1",(x+1,0),2*S,fontsize(9));
label("2",(x+2,0),2*S,fontsize(9));
label("3",(x+3,0),2*S,fontsize(9));
draw((x+0.5,0)--(x+0.5,-0.1));
draw((x+1,0)--(x+1,-0.1));
draw((x+1.5,0)--(x+1.5,-0.1));
draw((x+2,0)--(x+2,-0.1));
draw((x+2.5,0)--(x+2.5,-0.1));
draw((x+3,0)--(x+3,-0.1));
label("(d)",(x+1.5,-2),N);
/*The lines*/
pair K, L, M, N, O, P, Q, R;
K = (x+0,1);
L = (x+1,1);
M = (x+1,0.5);
N= (x+1.5,0.5);
O= (x+1.5, -0.5);
P = (x+2.5, -0.5);
Q = (x+2.5, 0.5);
R = (x+3, 0.5);
draw(K--L--M--N--O--P--Q--R);[/asy][asy]
size(150);
Label f;
f.p=fontsize(8);
xaxis(0,3);
yaxis(-2,2);
label("Time (s)",(3.2,0.03),N);
label(rotate(90)*"Acceleration",(-0.1,0),W);
label("$0$",(-0,0),SW,fontsize(9));
label("1",(1,0),2*S,fontsize(9));
label("2",(2,0),2*S,fontsize(9));
label("3",(3,0),2*S,fontsize(9));
draw((0.5,0)--(0.5,-0.1));
draw((1,0)--(1,-0.1));
draw((1.5,0)--(1.5,-0.1));
draw((2,0)--(2,-0.1));
draw((2.5,0)--(2.5,-0.1));
draw((3,0)--(3,-0.1));
label(rotate(90)*"Acceleration",(-0.1,0),W);
label("(e)",(1.5,-2),N);
/*The lines*/
pair A, B, C, D, E, F, G, H;
A = (0,1);
B = (1,1);
C = (1,0.5);
D = (1.5,0.5);
E = (1.5, -0.5);
F = (2.5, -0.5);
G = (2.5, 0.5);
H = (3, 0.5);
draw(A--B--C--D--E--F--G--H);
[/asy]
2008 F = Ma, 7
A toboggan sled is traveling at $\text{2.0 m/s}$ across the snow. The sled and its riders have a combined mass of $\text{120 kg}$. Another child ($m_{\text{child}} = \text{40 kg}$) headed in the opposite direction jumps on the sled from the front. She has a speed of $\text{5.0 m/s}$ immediately before she lands on the sled. What is the new speed of the sled? Neglect any effects of friction.
(a) $\text{0.25 m/s}$
(b) $\text{0.33 m/s}$
(c) $\text{2.75 m/s}$
(d) $\text{3.04 m/s}$
(e) $\text{3.67 m/s}$
2008 F = Ma, 10
Which is the best value for the mass of the block?
(a) $\text{3 kg}$
(b) $\text{5 kg}$
(c) $\text{10 kg}$
(d) $\text{20 kg}$
(e) $\text{30 kg}$
2008 F = Ma, 25
Two satellites are launched at a distance $R$ from a planet of negligible radius. Both satellites are launched in the tangential direction. The first satellite launches correctly at a speed $v_\text{0}$ and enters a circular orbit. The second satellite, however, is launched at a speed $\frac{1}{2}v_\text{0}$. What is the minimum distance between the second satellite and the planet over the course of its orbit?
(a) $\frac{1}{\sqrt{2}}R$
(b) $\frac{1}{2}R$
(c) $\frac{1}{3}R$
(d) $\frac{1}{4}R$
(e) $\frac{1}{7}R$
2008 AIME Problems, 4
There exist $ r$ unique nonnegative integers $ n_1 > n_2 > \cdots > n_r$ and $ r$ unique integers $ a_k$ ($ 1\le k\le r$) with each $ a_k$ either $ 1$ or $ \minus{} 1$ such that
\[ a_13^{n_1} \plus{} a_23^{n_2} \plus{} \cdots \plus{} a_r3^{n_r} \equal{} 2008.
\]Find $ n_1 \plus{} n_2 \plus{} \cdots \plus{} n_r$.
2008 F = Ma, 22
A bullet of mass $m_\text{1}$ strikes a pendulum of mass $m_\text{2}$ suspended from a pivot by a string of length $L$ with a horizontal velocity $v_\text{0}$. The collision is perfectly inelastic and the bullet sticks to the bob. Find the minimum velocity $v_\text{0}$ such that the bob (with the bullet inside) completes a circular vertical loop.
(a) $2\sqrt{Lg}$
(b) $\sqrt{5Lg}$
(c) $(m_\text{1}+m_\text{2})2\sqrt{Lg}/m_\text{1}$
(d) $(m_\text{1}-m_\text{2})\sqrt{Lg}/m_\text{2}$
(e) $(m_\text{1}+m_\text{2})\sqrt{5Lg}/m_\text{1}$
2008 F = Ma, 17
A mass $m$ is resting at equilibrium suspended from a vertical spring of natural length $L$ and spring constant $k$ inside a box as shown:
[asy]
//The Spring
import graph;
size(10cm);
guide coil(path g, real width=0.1, real margin = 1*width) {
real L = arclength(g);
real r = width / 2;
pair startpoint = arcpoint(g, margin);
real[][] isectiontimes = intersections(g, circle(c=startpoint,r=r));
real initialcirclecentertime = (isectiontimes.length == 1 ?
isectiontimes[0][0] : isectiontimes[1][0]);
pair startdir = dir(startpoint - point(g,initialcirclecentertime));
real startangle = atan2(startdir.y, startdir.x);
real startarctime = arclength(subpath(g, 0, initialcirclecentertime));
write(startarctime);
pair endpoint = arcpoint(g, L - margin);
real finalcirclecentertime = intersections(g, circle(c=endpoint,r=r))[0][0];
pair enddir = dir(endpoint - point(g,finalcirclecentertime));
real endangle = atan2(enddir.y, enddir.x);
real endarctime = arclength(subpath(g, 0, finalcirclecentertime));
write(endarctime);
real coillength = 2r;
real lengthalongcoils = L - 2*margin;
int numcoils = ceil(lengthalongcoils / coillength);
real anglesubtended = 2pi * numcoils - startangle + endangle;
real angleat(real arctime) {
return (arctime - startarctime) * (anglesubtended / (endarctime - startarctime)) + startangle;
}
pair f(real t) {
return arcpoint(g,t) + r * expi(angleat(t));
}
return subpath(g, 0, arctime(g, margin)) & graph(f, startarctime, endarctime, n=max(length(g), 20*numcoils+2), operator..) & subpath(g, arctime(g, L-margin), length(g));
}
draw(coil((0,0.25)--(0,1)));
//Outer Box
draw((-1,1)--(1,1),linewidth(2));
draw((-1,1)--(-1,-1.2),linewidth(2));
draw((-1,-1.2)--(1,-1.2),linewidth(2));
draw((1,1)--(1,-1.2),linewidth(2));
//Inner Box
draw((-0.2,0.25)--(0.2,0.25),linewidth(2));
path arc1=arc((-0.2,0.15),(-0.2,0.25),(-0.3,0.15));
path arc2=arc((0.2,0.15),(0.3,0.15),(0.2,0.25));
draw(arc1,linewidth(2));
draw(arc2,linewidth(2));
draw((-0.3,0.15)--(-0.3,-0.3),linewidth(2));
draw((0.3,0.15)--(0.3,-0.3),linewidth(2));
path arc3=arc((-0.2,-0.3),(-0.3,-0.3),(-0.2,-0.4));
draw(arc3,linewidth(2));
path arc4=arc((0.2,-0.3),(0.2,-0.4),(0.3,-0.3));
draw((-0.2,-0.4)--(0.2,-0.4),linewidth(2));
draw(arc4,linewidth(2));
[/asy]
The box begins accelerating upward with acceleration $a$. How much closer does the equilibrium position of the mass move to the bottom of the box?
(a) $(a/g)L$
(b) $(g/a)L$
(c) $m(g + a)/k$
(d) $m(g - a)/k$
(e) $ma/k$
2008 F = Ma, 14
A spaceborne energy storage device consists of two equal masses connected by a tether and rotating about their center of mass. Additional energy is stored by reeling in the tether; no external forces are applied. Initially the device has kinetic energy $E$ and rotates at angular velocity $\omega$. Energy is added until the device rotates at angular velocity $2\omega$. What is the new kinetic energy of the device?
(a) $\sqrt{2}E$
(b) $2E$
(c) $2\sqrt{2}E$
(d) $4E$
(e) $8E$
2008 F = Ma, 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) $\sqrt{3}\frac{EL}{\rho gh}$
2008 F = Ma, 3
The position [i]vs.[/i] time graph for an object moving in a straight line is shown below. What is the instantaneous velocity at $\text{t = 2 s}$?
[asy]
size(300);
// x-axis
draw((0,0)--(12,0));
// x-axis tick marks
draw((4,0)--(4,-0.2));
draw((8,0)--(8,-0.2));
draw((12,0)--(12,-0.2));
// x-axis labels
label("1",(4,0),2*S);
label("2",(8,0),2*S);
label("3",(12,0),2*S);
//y-axis
draw((0,-3)--(0,5));
//y-axis tick marks
draw((-0.2,-2)--(0,-2));
draw((-0.2,0)--(0,0));
draw((-0.2,2)--(0,2));
draw((-0.2,4)--(0,4));
// y-axis labels
label("-2",(0,-2),2*W);
label("0",(0,0),2*W);
label("2",(0,2),2*W);
label("4",(0,4),2*W);
// Axis Labels
label("Time (s)",(10,0),N);
label(rotate(90)*"Position (m)",(0,1),6*W);
//Line
draw((0,4)--(12,-2));
[/asy]
(a) $-\text{2 m/s}$
(b) $-\frac{1}{2} \ \text{m/s}$
(c) $\text{0 m/s}$
(d) $\text{2 m/s}$
(e) $\text{4 m/s}$
2008 F = Ma, 24
A ball is launched upward from the ground at an initial vertical speed of $v_\text{0}$ and begins bouncing vertically. Every time it rebounds, it loses a proportion of the magnitude of its velocity due to the inelastic nature of the collision, such that if the speed just before hitting the ground on a bounce is $v$, then the speed just after the bounce is $rv$, where $r < 1$ is a constant. Calculate the total length of time that the ball remains bouncing, assuming that any time associated with the actual contact of the ball with the ground is negligible.
(a) $\frac{2v_\text{0}}{g}\frac{1}{1-r}$
(b) $\frac{v_\text{0}}{g}\frac{r}{1-r}$
(c) $\frac{2v_\text{0}}{g}\frac{1-r}{r}$
(d) $\frac{2v_\text{0}}{g}\frac{1}{1-r^2}$
(e) $\frac{2v_\text{0}}{g}\frac{1}{1+(1-r)^2}$
2008 F = Ma, 12
A uniform disk rotates at a fixed angular velocity on an axis through its center normal to the plane of the disk, and has kinetic energy $E$. If the same disk rotates at the same angular velocity about an axis on the edge of the disk (still normal to the plane of the disk), what is its kinetic energy?
(a) $\frac{1}{2}E$
(b) $\frac{3}{2}E$
(c) $2E$
(d) $3E$
(e) $4E$
2008 F = Ma, 16
A [i]massless[/i] spring with spring constant $k$ is vertically mounted so that bottom end is firmly attached to the ground, and the top end free. A ball with mass $m$ falls vertically down on the top end of the spring, becoming attached, so that the ball oscillates vertically on the spring. What equation describes the acceleration a of the ball when it is at a height $y$ above the [i]original[/i] position of the top end of the spring? Let down be negative, and neglect air resistance; $g$ is the magnitude of the acceleration of free fall.
(a) $a=mv^2/y+g$
(b) $a=mv^2/k-g$
(c) $a=(k/m)y-g$
(d) $a=-(k/m)y+g$
(e) $a=-(k/m)y-g$
2008 F = Ma, 11
Which is the best value for the coefficient of friction between the block and the surface?
(a) $\text{0.05}$
(b) $\text{0.07}$
(c) $\text{0.09}$
(d) $\text{0.5}$
(e) $\text{0.6}$
2008 F = Ma, 18
A uniform circular ring of radius $R$ is fixed in place. A particle is placed on the axis of the ring at a distance much greater than $R$ and allowed to fall towards the ring under the influence of the ring’s gravity. The particle achieves a maximum speed $v$. The ring is replaced with one of the same (linear) mass density but radius $2R$, and the experiment is repeated. What is the new maximum speed of the particle?
(a) $\frac{1}{2}v$
(b) $\frac{1}{\sqrt{2}}v$
(c) $v$
(d) $\sqrt{2}v$
(e) $2v$
2008 F = Ma, 15
A uniform round tabletop of diameter $\text{4.0 m}$ and mass $\text{50.0 kg}$ rests on massless, evenly spaced legs of length $\text{1.0 m}$ and spacing $\text{3.0 m}$. A carpenter sits on the edge of the table. What is the maximum mass of the carpenter such that the table remains upright? Assume that the force exerted by the carpenter on the table is vertical and at the edge of the table.
[asy]
size(10cm);
import graph;
xaxis(-3.6,3.6);
//The legs
draw((-1.4,1.4)--(-1.4,0));
draw((-1.6,1.4)--(-1.6,0));
draw((1.4,1.4)--(1.4,0));
draw((1.6,1.4)--(1.6,0));
//The tabletop
draw((-2.5,1.6)--(-2.5,1.4)--(2.5,1.4)--(2.5,1.6)--cycle);
path a = ((-3.2,0.1)--(-3.2,1.5));
draw(a,Arrows);
label("1.0m",a,E);
draw((-2.6,1.6)--(-3.6,1.6),dashed+linewidth(0.4));
path c = (-1.4,-0.5)--(1.4,-0.5);
draw(c,Arrows);
label("3.0m",c,2*S);
draw((-1.5,-0.1)--(-1.5,-0.8),dashed+linewidth(0.4));
draw((1.5,-0.1)--(1.5,-0.8),dashed+linewidth(0.4));
draw((-2.5,-0.1)--(-2.5,-1.8),dashed+linewidth(0.4));
draw((2.5,-0.1)--(2.5,-1.8),dashed+linewidth(0.4));
path d = (-2.4,-1.5)--(2.4,-1.5);
draw(d,Arrows);
label("4.0m",d,2*S);
//The Man
pair A = (2.4,1.7);
draw(circle((2.5,3.7),0.5),linewidth(1.6));
draw((2.4,1.7)--(2.5,3.2),linewidth(1.6));
draw((2.15,2.8)--(2.85,2.8),linewidth(1.6));
draw(A--A+(0.6,0)--A+(0.4,-0.8)--A+(0.6,-1.2),linewidth(1.6));
draw(A--(A+(0.64,-0.12)--A+(0.73,-0.9)--A+(1,-0.9)),linewidth(1.6));
draw((2.15,2.8)--(2.15,2.4)--(2.9,2),linewidth(1.6));
draw((2.85,2.8)--(3,2.3)--(3.5,2.1),linewidth(1.6));
[/asy]
(a) $\text{67 kg}$
(b) $\text{75 kg}$
(c) $\text{81 kg}$
(d) $\text{150 kg}$
(e) $\text{350 kg}$
2008 F = Ma, 13
A mass is attached to the wall by a spring of constant $k$. When the spring is at its natural length, the mass is given a certain initial velocity, resulting in oscillations of amplitude $A$. If the spring is replaced by a spring of constant $2k$, and the mass is given the same initial velocity, what is the amplitude of the resulting oscillation?
(a) $\frac{1}{2}A$
(b) $\frac{1}{\sqrt{2}}A$
(c) $\sqrt{2}A$
(d) $2A$
(e) $4A$
2008 F = Ma, 19
A car has an engine which delivers a constant power. It accelerates from rest at time $t = 0$, and at $t = t_\text{0}$ its acceleration is $a_\text{0}$. What is its acceleration at $t = 2t_\text{0}$? Ignore energy loss due to friction.
(a) $\frac{1}{2}a_\text{0}$
(b) $\frac{1}{\sqrt{2}}a_\text{0}$
(c) $a_\text{0}$
(d) $\sqrt{2}a_\text{0}$
(e) $2a_\text{0}$
2008 F = Ma, 9
A ball of mass $m_\text{1}$ travels along the x-axis in the positive direction with an initial speed of $v_{\text{0}}$. It collides with a ball of mass $m_\text{2}$ that is originally at rest. After the collision, the ball of mass $m_\text{1}$ has velocity $v_{\text{1x}}\hat{x}+v_{\text{1y}}\hat{y}$ and the ball of mass $m_\text{2}$ has velocity $v_{\text{2x}}\hat{x}+v_{\text{2y}}\hat{y}$.
Consider the following five statements:
$\text{I)} \ \ \ \ \ \ 0=m_{\text{1}}v_{\text{1x}}+m_{\text{1}}v_{\text{2x}}$
$\text{II)} \ \ \ \ \ m_{\text{1}}v_{\text{0}}=m_{\text{1}}v_{\text{1y}}+m_{\text{2}}v_{\text{2y}}$
$\text{III)} \ \ \ \ 0=m_{\text{1}}v_{\text{1y}}+m_{\text{2}}v_{\text{2y}}$
$\text{IV)} \ \ \ \ m_{\text{1}}v_{\text{0}}=m_{\text{1}}v_{\text{1x}}+m_{\text{1}}v_{\text{1y}}$
$\text{V)} \ \ \ \ \ m_{\text{1}}v_{\text{0}}=m_{\text{1}}v_{\text{1x}}+m_{\text{2}}v_{\text{2x}}$
Of these five statements, the system must satisfy
(a) $\text{I and II}$
(b) $\text{III and V}$
(c) $\text{II and V}$
(d) $\text{III and IV}$
(e) $\text{I and III}$
2008 F = Ma, 8
Riders in a carnival ride stand with their backs against the wall of a circular room of diameter $\text{8.0 m}$. The room is spinning horizontally about an axis through its center at a rate of $\text{45 rev/min}$ when the floor drops so that it no longer provides any support for the riders. What is the minimum coefficient of static friction between the wall and the rider required so that the rider does not slide down the wall?
(a) $\text{0.0012}$
(b) $\text{0.056}$
(c) $\text{0.11}$
(d) $\text{0.53}$
(e) $\text{8.9}$
2008 F = Ma, 1
A bird flying in a straight line, initially at $\text{10 m/s}$, uniformly increases its speed to $\text{18 m/s}$ while covering a distance of $\text{40 m}$. What is the magnitude of the acceleration of the bird?
(a) $\text{0.1 m/s}^2$
(b) $\text{0.2 m/s}^2$
(c) $\text{2.0 m/s}^2$
(d) $\text{2.8 m/s}^2$
(e) $\text{5.6 m/s}^2$
2008 F = Ma, 21
Consider a particle at rest which may decay into two (daughter) particles or into three (daughter) particles. Which of the following is true in the two-body case but false in the three-body case? (There are no external forces.)
(a) The velocity vectors of the daughter particles must lie in a single plane.
(b) Given the total kinetic energy of the system and the mass of each daughter particle, it is possible to determine the speed of each daughter particle.
(c) Given the speed(s) of all but one daughter particle, it is possible to determine the speed of the remaining particle.
(d) The total momentum of the daughter particles is zero.
(e) None of the above.