Found problems: 4
2011 F = Ma, 5
A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $\text{150,000,000 km}$ from the Sun. Which of the following is the closest for the acceleration of the Earth in this orbit?
(A) $\text{exactly 0 m/s}^2$
(B) $\text{0.006 m/s}^2$
(C) $\text{0.6 m/s}^2$
(D) $\text{6 m/s}^2$
(E) $\text{10 m/s}^2$
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]
2010 F = Ma, 5
Two projectiles are launched from a $35$ meter ledge as shown in the diagram. One is launched from a $37$ degree angle above the horizontal and the other is launched from $37$ degrees below the horizontal. Both of the launches are given the same initial speed of $v_\text{0} = \text{50 m/s}$.
[asy]
size(300);
import graph;
draw((-8,0)--(0,0)--(0,-11)--(30,-11));
draw((0,-11)--(-4.5,-11),dashdotted);
draw((0,0)--(12,0),dashdotted);
label(scale(0.75)*"35 m",(0,-5.5),5*W);
draw((-4,-4.5)--(-4,-0.5),EndArrow(size=5));
draw((-4,-6)--(-4,-10.5),EndArrow(size=5));
// Projectiles
real f(real x){ return -11x^2/49; }
draw(graph(f,0,7),dashed+linewidth(1.5));
real g(real x){ return -6x^2/145+119x/145; }
draw(graph(g,0,29),dashed+linewidth(1.5));
// Labels
label(scale(0.75)*"Projectile 1",(20,2),E);
label(scale(0.75)*"Projectile 2",(6,-7),E);
[/asy]
The difference in the times of flight for these two projectiles, $t_1-t_2$, is closest to
(A) $\text{3 s}$
(B) $\text{5 s}$
(C) $\text{6 s}$
(D) $\text{8 s}$
(E) $\text{10 s}$
2009 F = Ma, 5
Three equal mass satellites $A$, $B$, and $C$ are in coplanar orbits around a planet as shown in the figure. The magnitudes of the angular momenta of the satellites as measured about the planet are $L_A$, $L_B$, and $L_C$. Which of the following statements is correct?
[asy]
// Code created by riben
size(250);
dotfactor=12;
draw(circle((0,0),1.5),linewidth(2));
draw(circle((0,0),6),dashdotted);
draw(circle((0,0),14),dashed);
draw(ellipse((4,0),10,8),linewidth(1));
pair A,B,C;
A=(-7,12.12);
B=(5,7.9);
C=(5.7,-1.87);
dot(A);
dot(B);
dot(C);
label("A",A,NW*1.5);
label("B",B,NW*1.5);
label("C",C,E*1.5);
filldraw((-1.500, 0.078)--
(-1.428, 0.080)--
(-1.337, 0.094)--
(-1.295, 0.157)--
(-1.246, 0.209)--
(-1.186, 0.227)--
(-1.143, 0.290)--
(-1.148, 0.357)--
(-1.135, 0.469)--
(-1.057, 0.505)--
(-0.996, 0.563)--
(-0.936, 0.526)--
(-0.852, 0.557)--
(-0.773, 0.587)--
(-0.772, 0.716)--
(-0.765, 0.828)--
(-0.781, 0.955)--
(-0.732, 1.035)--
(-0.648, 1.083)--
(-0.605, 1.162)--
(-0.604, 1.246)--
(-0.645, 1.295)--
(-0.736, 1.270)--
(-0.796, 1.229)--
(-0.851, 1.193)--
(-0.941, 1.135)--
(-1.014, 1.076)--
(-1.105, 0.995)--
(-1.154, 0.921)--
(-1.227, 0.841)--
(-1.288, 0.760)--
(-1.349, 0.669)--
(-1.398, 0.556)--
(-1.453, 0.465)--
(-1.485, 0.357)--
(-1.510, 0.239)--cycle,gray);
filldraw((-0.119, 1.245)--
(-0.130, 1.193)--
(-0.146, 1.095)--
(-0.202, 1.056)--
(-0.327, 1.033)--
(-0.262, 1.031)--
(-0.278, 0.979)--
(-0.193, 0.949)--
(-0.108, 0.943)--
(-0.013, 0.941)--
(0.032, 0.915)--
(0.026, 0.840)--
(0.015, 0.779)--
(0.019, 0.705)--
(0.074, 0.646)--
(0.113, 0.582)--
(0.162, 0.533)--
(0.167, 0.463)--
(0.241, 0.400)--
(0.311, 0.412)--
(0.416, 0.410)--
(0.465, 0.342)--
(0.541, 0.410)--
(0.611, 0.347)--
(0.679, 0.242)--
(0.728, 0.132)--
(0.732, 0.048)--
(0.671, -0.037)--
(0.615, -0.104)--
(0.540, -0.172)--
(0.409, -0.209)--
(0.324, -0.244)--
(0.253, -0.293)--
(0.188, -0.314)--
(0.162, -0.389)--
(0.181, -0.486)--
(0.270, -0.534)--
(0.340, -0.537)--
(0.380, -0.596)--
(0.424, -0.688)--
(0.418, -0.772)--
(0.352, -0.825)--
(0.281, -0.883)--
(0.241, -0.926)--
(0.145, -0.981)--
(0.044, -1.044)--
(-0.006, -1.107)--
(-0.007, -1.190)--
(0.077, -1.216)--
(0.162, -1.213)--
(0.253, -1.163)--
(0.323, -1.128)--
(0.404, -1.075)--
(0.510, -1.015)--
(0.605, -0.980)--
(0.671, -0.931)--
(0.731, -0.920)--
(0.817, -0.852)--
(0.898, -0.798)--
(0.963, -0.777)--
(0.964, -0.708)--
(1.024, -0.645)--
(1.025, -0.571)--
(0.976, -0.488)--
(0.912, -0.425)--
(0.878, -0.347)--
(0.823, -0.289)--
(0.779, -0.225)--
(0.744, -0.193)--
(0.756, -0.100)--
(0.816, -0.033)--
(0.837, 0.047)--
(0.838, 0.122)--
(0.824, 0.200)--
(0.800, 0.307)--
(0.796, 0.381)--
(0.872, 0.416)--
(0.967, 0.414)--
(1.016, 0.360)--
(1.096, 0.381)--
(1.117, 0.428)--
(1.058, 0.506)--
(0.998, 0.564)--
(0.954, 0.591)--
(0.914, 0.617)--
(0.860, 0.676)--
(0.800, 0.716)--
(0.751, 0.775)--
(0.757, 0.859)--
(0.797, 0.921)--
(0.823, 0.987)--
(0.889, 1.096)--
(0.850, 1.160)--
(0.780, 1.176)--
(0.700, 1.183)--
(0.645, 1.125)--
(0.579, 1.039)--
(0.518, 0.986)--
(0.438, 0.956)--
(0.343, 0.967)--
(0.289, 1.049)--
(0.249, 1.117)--
(0.195, 1.176)--
(0.125, 1.192)--
(0.030, 1.208)--
(-0.040, 1.220)--cycle,gray);
[/asy]
(A) $L_\text{A} > L_\text{B} > L_\text{C}$
(B) $L_\text{C} > L_\text{B} > L_\text{A}$
(C) $L_\text{B} > L_\text{C} > L_\text{A}$
(D) $L_\text{B} > L_\text{A} > L_\text{C}$
(E) The relationship between the magnitudes is different at various instants in time.