{"id":1407,"date":"2018-02-06T15:47:34","date_gmt":"2018-02-06T15:47:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/?post_type=chapter&p=1407"},"modified":"2018-02-06T15:47:34","modified_gmt":"2018-02-06T15:47:34","slug":"4-chapter-review","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/4-chapter-review\/","title":{"raw":"4 Chapter Review","rendered":"4 Chapter Review"},"content":{"raw":"
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Key Terms<\/span><\/h3>\r\n
\r\n \t
acceleration vector<\/strong><\/dt>\r\n \t
instantaneous acceleration found by taking the derivative of the velocity function with respect to time in unit vector notation<\/dd>\r\n<\/dl>\r\n
\r\n \t
angular frequency<\/strong><\/dt>\r\n \t
\u03c9<\/span>,<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9,<\/span><\/span>\u00a0rate of change of an angle with which an object that is moving on a circular path<\/dd>\r\n<\/dl>\r\n
\r\n \t
centripetal acceleration<\/strong><\/dt>\r\n \t
component of acceleration of an object moving in a circle that is directed radially inward toward the center of the circle<\/dd>\r\n<\/dl>\r\n
\r\n \t
displacement vector<\/strong><\/dt>\r\n \t
vector from the initial position to a final position on a trajectory of a particle<\/dd>\r\n<\/dl>\r\n
\r\n \t
position vector<\/strong><\/dt>\r\n \t
vector from the origin of a chosen coordinate system to the position of a particle in two- or three-dimensional space<\/dd>\r\n<\/dl>\r\n
\r\n \t
projectile motion<\/strong><\/dt>\r\n \t
motion of an object subject only to the acceleration of gravity<\/dd>\r\n<\/dl>\r\n
\r\n \t
range<\/strong><\/dt>\r\n \t
maximum horizontal distance a projectile travels<\/dd>\r\n<\/dl>\r\n
\r\n \t
reference frame<\/strong><\/dt>\r\n \t
coordinate system in which the position, velocity, and acceleration of an object at rest or moving is measured<\/dd>\r\n<\/dl>\r\n
\r\n \t
relative velocity<\/strong><\/dt>\r\n \t
velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to another reference frame<\/dd>\r\n<\/dl>\r\n
\r\n \t
tangential acceleration<\/strong><\/dt>\r\n \t
magnitude of which is the time rate of change of speed. Its direction is tangent to the circle.<\/dd>\r\n<\/dl>\r\n
\r\n \t
time of flight<\/strong><\/dt>\r\n \t
elapsed time a projectile is in the air<\/dd>\r\n<\/dl>\r\n
\r\n \t
total acceleration<\/strong><\/dt>\r\n \t
vector sum of centripetal and tangential accelerations<\/dd>\r\n<\/dl>\r\n
\r\n \t
trajectory<\/strong><\/dt>\r\n \t
path of a projectile through the air<\/dd>\r\n<\/dl>\r\n
\r\n \t
velocity vector<\/strong><\/dt>\r\n \t
vector that gives the instantaneous speed and direction of a particle; tangent to the trajectory<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/div>\r\n
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Key Equations<\/h3>\r\n<\/div>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Position vector<\/td>\r\nr<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>x<\/span>(<\/span>t<\/span>)<\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>y<\/span>(<\/span>t<\/span>)<\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>z<\/span>(<\/span>t<\/span>)<\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r\u2192(t)=x(t)i^+y(t)j^+z(t)k^<\/span><\/span><\/td>\r\n<\/tr>\r\n
Displacement vector<\/td>\r\n\u0394<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>2<\/span><\/span>)<\/span>\u2212<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>1<\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u0394r\u2192=r\u2192(t2)\u2212r\u2192(t1)<\/span><\/span><\/td>\r\n<\/tr>\r\n
Velocity vector<\/td>\r\nv<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>lim<\/span><\/span>\u0394<\/span>t<\/span>\u2192<\/span>0<\/span><\/span><\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>\u0394<\/span>t<\/span>)<\/span>\u2212<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span>\u0394<\/span>t<\/span><\/span><\/span>=<\/span>d<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v\u2192(t)=lim\u0394t\u21920r\u2192(t+\u0394t)\u2212r\u2192(t)\u0394t=dr\u2192dt<\/span><\/span><\/td>\r\n<\/tr>\r\n
Velocity in terms of components<\/td>\r\nv<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>v<\/span>x<\/span><\/span>(<\/span>t<\/span>)<\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>v<\/span>y<\/span><\/span>(<\/span>t<\/span>)<\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>v<\/span>z<\/span><\/span>(<\/span>t<\/span>)<\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v\u2192(t)=vx(t)i^+vy(t)j^+vz(t)k^<\/span><\/span><\/td>\r\n<\/tr>\r\n
Velocity components<\/td>\r\nv<\/span>x<\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>d<\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span>v<\/span>y<\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>d<\/span>y<\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span>v<\/span>z<\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>d<\/span>z<\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>vx(t)=dx(t)dtvy(t)=dy(t)dtvz(t)=dz(t)dt<\/span><\/span><\/td>\r\n<\/tr>\r\n
Average velocity<\/td>\r\nv<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>avg<\/span><\/span><\/span>=<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>2<\/span><\/span>)<\/span>\u2212<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>1<\/span><\/span>)<\/span><\/span>t<\/span>2<\/span><\/span>\u2212<\/span>t<\/span>1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v\u2192avg=r\u2192(t2)\u2212r\u2192(t1)t2\u2212t1<\/span><\/span><\/td>\r\n<\/tr>\r\n
Instantaneous acceleration<\/td>\r\na<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>lim<\/span><\/span>t<\/span>\u2192<\/span>0<\/span><\/span><\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>+<\/span>\u0394<\/span>t<\/span>)<\/span>\u2212<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span>\u0394<\/span>t<\/span><\/span><\/span>=<\/span>d<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a\u2192(t)=limt\u21920v\u2192(t+\u0394t)\u2212v\u2192(t)\u0394t=dv\u2192(t)dt<\/span><\/span><\/td>\r\n<\/tr>\r\n
Instantaneous acceleration, component form<\/td>\r\na<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>d<\/span>v<\/span>x<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span><\/span><\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>d<\/span>v<\/span>y<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span><\/span><\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>d<\/span>v<\/span>z<\/span><\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span><\/span><\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a\u2192(t)=dvx(t)dti^+dvy(t)dtj^+dvz(t)dtk^<\/span><\/span><\/td>\r\n<\/tr>\r\n
Instantaneous acceleration as second\r\n
<\/div>\r\nderivatives of position<\/td>\r\n
a<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>d<\/span>2<\/span><\/span>x<\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span>2<\/span><\/span><\/span><\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>d<\/span>2<\/span><\/span>y<\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span>2<\/span><\/span><\/span><\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>d<\/span>2<\/span><\/span>z<\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span>2<\/span><\/span><\/span><\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a\u2192(t)=d2x(t)dt2i^+d2y(t)dt2j^+d2z(t)dt2k^<\/span><\/span><\/td>\r\n<\/tr>\r\n
Time of flight<\/td>\r\nT<\/span>tof<\/span><\/span><\/span>=<\/span>2<\/span>(<\/span>v<\/span>0<\/span><\/span>sin<\/span>\u03b8<\/span>)<\/span><\/span>g<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Ttof=2(v0sin\u03b8)g<\/span><\/span><\/td>\r\n<\/tr>\r\n
Trajectory<\/td>\r\ny<\/span>=<\/span>(<\/span>tan<\/span>\u03b8<\/span>0<\/span><\/span>)<\/span>x<\/span>\u2212<\/span>[<\/span>g<\/span>2<\/span>(<\/span>v<\/span>0<\/span><\/span>cos<\/span>\u03b8<\/span>0<\/span><\/span>)<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>]<\/span><\/span>x<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>y=(tan\u03b80)x\u2212[g2(v0cos\u03b80)2]x2<\/span><\/span><\/td>\r\n<\/tr>\r\n
Range<\/td>\r\nR<\/span>=<\/span>v<\/span>2<\/span>0<\/span><\/span>sin<\/span><\/span>2<\/span>\u03b8<\/span>0<\/span><\/span><\/span>g<\/span><\/span><\/span><\/span><\/span><\/span><\/span>R=v02sin2\u03b80g<\/span><\/span><\/td>\r\n<\/tr>\r\n
Centripetal acceleration<\/td>\r\na<\/span>C<\/span><\/span>=<\/span>v<\/span>2<\/span><\/span><\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>aC=v2r<\/span><\/span><\/td>\r\n<\/tr>\r\n
Position vector, uniform circular motion<\/td>\r\nr<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>A<\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span><\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>A<\/span><\/span>sin<\/span><\/span>\u03c9<\/span>t<\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r\u2192(t)=Acos\u03c9ti^+Asin\u03c9tj^<\/span><\/span><\/td>\r\n<\/tr>\r\n
Velocity vector, uniform circular motion<\/td>\r\nv<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>d<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span><\/span><\/span>=<\/span>\u2212<\/span>A<\/span>\u03c9<\/span><\/span>sin<\/span><\/span>\u03c9<\/span>t<\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>A<\/span>\u03c9<\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v\u2192(t)=dr\u2192(t)dt=\u2212A\u03c9sin\u03c9ti^+A\u03c9cos\u03c9tj^<\/span><\/span><\/td>\r\n<\/tr>\r\n
Acceleration vector, uniform circular motion<\/td>\r\na<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span>=<\/span>d<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>(<\/span>t<\/span>)<\/span><\/span>d<\/span>t<\/span><\/span><\/span>=<\/span>\u2212<\/span>A<\/span>\u03c9<\/span>2<\/span><\/span><\/span>cos<\/span><\/span>\u03c9<\/span>t<\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>\u2212<\/span>A<\/span>\u03c9<\/span>2<\/span><\/span><\/span>sin<\/span><\/span>\u03c9<\/span>t<\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a\u2192(t)=dv\u2192(t)dt=\u2212A\u03c92cos\u03c9ti^\u2212A\u03c92sin\u03c9tj^<\/span><\/span><\/td>\r\n<\/tr>\r\n
Tangential acceleration<\/td>\r\na<\/span>T<\/span><\/span>=<\/span>d<\/span>\u2223\u2223<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u2223\u2223<\/span><\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>aT=d|v\u2192|dt<\/span><\/span><\/td>\r\n<\/tr>\r\n
Total acceleration<\/td>\r\na<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>a<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>C<\/span><\/span>+<\/span>a<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>T<\/span><\/span><\/span><\/span><\/span><\/span><\/span>a\u2192=a\u2192C+a\u2192T<\/span><\/span><\/td>\r\n<\/tr>\r\n
Position vector in frame\r\n
<\/div>\r\nS<\/em>\u00a0is the position\r\n
<\/div>\r\nvector in frame\u00a0S<\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span>S\u2032<\/span><\/span>\u00a0plus the vector from the\r\n
<\/div>\r\norigin of\u00a0S<\/em>\u00a0to the origin of\u00a0S<\/span>\u2032<\/span><\/span><\/span><\/span><\/span><\/span><\/span>S\u2032<\/span><\/span><\/td>\r\n
r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>P<\/span>S<\/span><\/span><\/span>=<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>P<\/span>S<\/span>\u2032<\/span><\/span><\/span><\/span>+<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>S<\/span>\u2032<\/span><\/span>S<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>r\u2192PS=r\u2192PS\u2032+r\u2192S\u2032S<\/span><\/span><\/td>\r\n<\/tr>\r\n
Relative velocity equation connecting two\r\n
<\/div>\r\nreference frames<\/td>\r\n
v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>P<\/span>S<\/span><\/span><\/span>=<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>P<\/span>S<\/span>\u2032<\/span><\/span><\/span><\/span>+<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>S<\/span>\u2032<\/span><\/span>S<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v\u2192PS=v\u2192PS\u2032+v\u2192S\u2032S<\/span><\/span><\/td>\r\n<\/tr>\r\n
Relative velocity equation connecting more\r\n
<\/div>\r\nthan two reference frames<\/td>\r\n
v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>P<\/span>C<\/span><\/span><\/span>=<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>P<\/span>A<\/span><\/span><\/span>+<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>A<\/span>B<\/span><\/span><\/span>+<\/span>v<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>B<\/span>C<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>v\u2192PC=v\u2192PA+v\u2192AB+v\u2192BC<\/span><\/span><\/td>\r\n<\/tr>\r\n
Relative acceleration equation<\/td>\r\na<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>P<\/span>S<\/span><\/span><\/span>=<\/span>a<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>P<\/span>S<\/span>\u2032<\/span><\/span><\/span><\/span>+<\/span>a<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>S<\/span>\u2032<\/span><\/span>S<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a\u2192PS=a\u2192PS\u2032+a\u2192S\u2032S<\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n
<\/div>\r\n
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Summary<\/span><\/h3>\r\n
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4.1<\/span>\u00a0<\/span>Displacement and Velocity Vectors<\/span><\/h4>\r\n