Post on 14-Jul-2020
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Physical Sciences
Physical Sciences X-Sheets
Physical Sciences
VERTICAL PROJECTILE MOTION
KEY CONCEPTSThis lesson focuses on the following key concepts:
• Forceofgravity
• Gravitationalacceleration
• Equationsofmotion
TERMINOLOGY & DEFINITIONSAcceleration due to gravity:istheaccelerationexperiencedbyaprojectileduetothegravitational
forcethatactsonit
Gravitational force:istheforceofattractionbetweenanytwoobjectsintheuniverse.Inthecase
ofanobjectonearth,itwillbetheforcewithwhichtheearth,asthebiggerobject,willattractthe
objecttowardsitscentre.Thisgravitationalforceiscalledtheweightofanobjectandismeasured
in Newton (N)
Height:theverticaldisplacementofanobject
X-PLANATION OF KEY CONCEPTS AND TERMINOLOGIES
The Gravitational ForceThegravitationalforceactingontheobjectduringitsmotionisfoundbyusingtheformula:
Fg = ma where: F
gisthegravitationalforceactingontheobject
misthemassoftheobjectand aisaccelerationduetogravity
Thegravitationalforceactingontheprojectileisactuallytheforceofattractionbetweentheearthandtheprojectile.Thustheformulaonthelawofuniversalgravityapplies.Therefore,theforceexperiencedbyaprojectileistheforceofattractionbetweentheearthandtheprojectilewhich
canbefoundusingthefollowingformula:
Fg =
where:Fisthegravitationalforcebetweentheearthandtheprojectile
Misthemassoftheearthand misthemassoftheprojectile
It,therefore,followsthat ma = G .Fromthisequation,itcanbeseenthat
accelerationduetogravity,a = G .Thisequationtellsthattheaccelerationdueto
gravitydoesnotdependonthemassofanobject,butonlyonthemassoftheearthandthedistancebetweentheearthandtheobject.
Mm
r2
Mm
r2
Mm
r2
VerticalProjectileMotion
Physical Sciences X-Sheets
15
Projectile motionVerticalprojectilemotionismotionofanobjectinanupordowndirection.Anobjectinanupordownmotioniscalledaprojectile.Theonlyforcethatactsonaprojectileistheforceduetogravity(gravitationalforce).Thisforceequalstotheweightoftheobject.Sincethemovementofaprojectileisinupordownmotion,itsdisplacementisinaverticaldirectionand,therefore,thesymbolusedforitisΔy.Allprojectilesexperiencethesameaccelerationduringmotion.Thisaccelerationisaccelerationduetogravity,whichis9.8m∙s-2inadownwarddirection(towardsthecentreoftheearth).
Whenanobjectmovesinanupordownmotion,itissaidtobemovinginonedimension.Thefollowingequa-tionsareusedinsolvingverticalprojectilemotionproblems:
Notice that ∆y isusedforthedisplacementsincetheobjectmovesinaverticaldirection.
Graphs of projectile motionTheaccelerationduetogravityforaprojectileisconstantandequaltog=9.8m∙s-2downwards.Thisapplieswhenaprojectileismovingupordown.Wemustassignadirectiontomotion.Itisusualtomakeuppositiveand,therefore,g=-9,8m.s-2Theaccelerationversustimegraphforthismotionisalwaysthesame
asshownbelow:
Astoneisthrownupwardsfromaparticularheighthabovetheground.Asketchgraphofthepositionversus
timeforthemotionofthestoneisdrawnbelow.
Vf = Vi + a∆t V2f = V2
i + 2a∆y
∆y = Vi∆t + a∆t2 12 ∆y = ∆t ( )V
f + V
i
2
VERTICAL PROJECTILE MOTION
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Physical Sciences
Physical Sciences X-Sheets
Physical Sciences
Notice from the graph that the time the stone takes to reach its maximum height above the ground, Δt1, equals Δt2, the time it takes to fall from maximum height and return to the point of projection, h. The graph below is the velocity versus time graph of the stone thrown up from h.
Notice from the graph that when the stone reaches its maximum height, its velocity becomes zero. Also notice that the velocity with which the stone reaches the point of projection equals the velocity with which it was projected. However, the two velocities are in opposing directions. Thus if the velocity of the stone in the upwards direction is taken to be positive, its velocity as it falls back to its point of projection will be negative as shown in the graph.
X-AMPLE 1Anobjectisthrownupwardswithavelocityof24,5m∙s-1andtakes5storeturntothethrower’shand.Usetheinformationtosketchthefollowinggraphs:
a) Displacementvstime
b) Velocityvstime
c) Accelerationvstime
X-AMPLE 2Aboystandingontopofabuildingthrowsaballverticallyupwardsfromaposition,3,5mabovethe
groundwithaninitialvelocityof10m∙s-1.Ignoretheeffectsofairresistanceandanswerthe
followingquestions.
2.1 Whatisthemagnitudeanddirectionoftheaccelerationoftheball.
2.2 Calculatethemaximumheightreachedbytheballabovetheground.
2.3 Whatwasthevelocityoftheballatitsmaximumheight?
2.4 Calculatethetimetakenbytheballtoreachitsmaximumheight.
2.5 Howmuchtimedidtheballtaketoreachitsmaximumpositionandbacktothe
positionfromwhichitwasthrown.
2.6 Calculatethetotaltimetakenbytheballtoreachtheground.
2.7 Calculatethevelocitywithwhichtheballhitstheground.
2.8 Drawaroughsketchofthevelocity-timegraph.Showrelevantpointsonyour
velocityandtimeaxis.
VERTICAL PROJECTILE MOTION
Physical Sciences X-Sheets
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X-ERCISE1.Aboystandsattheedgeofahighcliff.Hethrowsastoneverticallyupwardswithaninitialvelocityof
9,8m∙s-1.Thestonestrikesthegroundatapointbelowthecliffafter3,5seconds.Thevelocity-timegraph
belowwasobtainedfrommeasurementsmadeduringthemotionofthestone.
GRAPH OF VELOCITY VS TIME
Usetheinformationonthegraphtoanswerthefollowingquestions:
1.1 Calculatetheaccelerationofthestonebetweentimest=2sandt=3s.
1.2 Atwhichtime(s)isthestonemovingataspeedof5m∙s-1?
1.3 Afterhowmanysecondsdoesthestonereachitshighestpoint?
1.4 Determinetheheightoftheclifffromwhichthestonewasthrown.
1.5 Usingthetopofthecliffastheinitialpositionofthestone,sketchtheposition-timegraph
formotionofthestonefromitshighestpointuntilitreachestheground.Onlyindicate
relevanttimevaluesonthex–axis.
VERTICAL PROJECTILE MOTION
Vel
ocit
y (m
.s-1)
Time(s)
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Physical Sciences X-Sheets
Physical Sciences
ANSWERS 1.1 10m∙s-2downwards(Note:graphisnotaccurateenoughtoread9,8m.s-2)
1.2 Time:0,5secondsafterbeingthrown;1,5secondsafterbeingthrown
1.3 Afteronesecond
1.4 31,25m
1.5
VERTICAL PROJECTILE MOTION