Chapter 4 – 2D and 3D MotionChapter 4 – 2D and 3D Motion
I.I. DefinitionsDefinitions
II.II. Projectile motionProjectile motion
III.III. Uniform circular motionUniform circular motion
IV.IV. Relative motionRelative motion
Motion in Two DimensionsMotion in Two Dimensions
• Using Using ++ or or –– signs is not always sufficient to fully signs is not always sufficient to fully describe motion in more than one dimensiondescribe motion in more than one dimension Vectors can be used to more fully describe motionVectors can be used to more fully describe motion
• Still interested in displacement, velocity, and Still interested in displacement, velocity, and accelerationacceleration
• Will serve as the basis of multiple types of Will serve as the basis of multiple types of motion in future chaptersmotion in future chapters
Position and DisplacementPosition and Displacement
• The position of an The position of an object is described by object is described by its position vector,its position vector, rr
• The The displacementdisplacement of of the object is defined the object is defined as the as the change in its change in its positionposition
Δr = rΔr = rff - r - rii
General Motion IdeasGeneral Motion Ideas
• In two- or three-dimensional kinematics, In two- or three-dimensional kinematics, everything is the same as in one-everything is the same as in one-dimensional motion except that we must dimensional motion except that we must now use full vector notationnow use full vector notation
Positive and negative signs are no longer Positive and negative signs are no longer sufficient to determine the directionsufficient to determine the direction
Position vector:Position vector: extends from the origin of a coordinate extends from the origin of a coordinate system to the particle.system to the particle.
kzjyixr ˆˆˆ
kzzjyyixxrrr ˆ)(ˆ)(ˆ)( 12121212
I.I. DefinitionsDefinitions
Displacement vector:Displacement vector: represents a particle’s position represents a particle’s position change during a certain time interval.change during a certain time interval.
Average VelocityAverage Velocity
• The average velocity is the The average velocity is the ratio of the displacement ratio of the displacement to the time interval for the to the time interval for the displacementdisplacement
The direction of the average The direction of the average velocity is the direction of velocity is the direction of
the displacement vector, the displacement vector, ΔrΔr
tv
r
Average VelocityAverage Velocity
• The average velocity between points is The average velocity between points is independent of the pathindependent of the path taken taken
This is because it is dependent on the This is because it is dependent on the displacement, also independent of the pathdisplacement, also independent of the path
I.I. DefinitionsDefinitions
Average velocityAverage velocity::
kt
zj
t
yit
x
t
rvavg ˆˆˆ
Instantaneous VelocityInstantaneous Velocity
• The instantaneous velocity is the limit of The instantaneous velocity is the limit of the average velocity as the average velocity as ΔΔtt approaches approaches zerozero
The direction of the instantaneous velocity is The direction of the instantaneous velocity is along a line that is tangent to the path of the along a line that is tangent to the path of the particle’s direction of motionparticle’s direction of motion
0limt
d
t dt
r r
v
Instantaneous velocityInstantaneous velocity:
kdt
dzj
dt
dyi
dt
dx
dt
rdkvjvivv zyx
ˆˆˆˆˆˆ
The direction of the instantaneous velocity of a particle is The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s positionalways tangent to the particle’s path at the particle’s position
Average AccelerationAverage Acceleration
• The average acceleration of a particle as it The average acceleration of a particle as it moves is defined as the change in the moves is defined as the change in the instantaneous velocity vector divided by instantaneous velocity vector divided by the time interval during which that change the time interval during which that change occurs.occurs.
f i
f it t t
v v v
a
Average AccelerationAverage Acceleration
• As a particle As a particle moves,moves, ΔvΔv can be can be found in different found in different waysways
• The average The average acceleration is a acceleration is a vector quantity vector quantity directed alongdirected along ΔvΔv
t
v
t
vvaavg
12
Instantaneous AccelerationInstantaneous Acceleration
• The instantaneous acceleration is the limit The instantaneous acceleration is the limit of the average acceleration as of the average acceleration as ΔΔtt approaches zeroapproaches zero
0limt
d
t dt
v v
a
Instantaneous acceleration:Instantaneous acceleration:
kdt
dvj
dt
dvi
dt
dv
dt
vdkajaiaa zyxzyx
ˆˆˆˆˆˆ
Producing An AccelerationProducing An Acceleration
• Various changes in a particle’s motion Various changes in a particle’s motion may produce an accelerationmay produce an acceleration– The magnitude of the velocity vector may The magnitude of the velocity vector may
changechange
– The direction of the velocity vector may The direction of the velocity vector may changechange
Even if the magnitude remains constantEven if the magnitude remains constant
– Both may change simultaneouslyBoth may change simultaneously
Kinematic Equations for Two-Kinematic Equations for Two-Dimensional MotionDimensional Motion
• When the two-dimensional motion has a When the two-dimensional motion has a constant acceleration, a series of constant acceleration, a series of equations can be developed that describe equations can be developed that describe the motionthe motion
• These equations will be similar to those of These equations will be similar to those of one-dimensional kinematicsone-dimensional kinematics
Kinematic EquationsKinematic Equations
• Position vectorPosition vector
• VelocityVelocity
• Since acceleration is constant, we can also Since acceleration is constant, we can also find an expression for the velocity as a find an expression for the velocity as a function of time: function of time:
vvff = = vvii + + aatt
ˆ ˆx y r i j
ˆ ˆx y
dv v
dt
rv i j
Kinematic EquationsKinematic Equations
• The velocity vector The velocity vector can be represented can be represented by its componentsby its components
vvff is generally not is generally not
along the direction of along the direction of eithereither vvii oror aatt
Kinematic EquationsKinematic Equations• The position vector can also The position vector can also
be expressed as a function of be expressed as a function of time:time:
rrff = r = rii + v + viitt + ½ a + ½ att22
– This indicates that the This indicates that the position vector is the sum position vector is the sum of three other vectors:of three other vectors:• The initial position vectorThe initial position vector• TheThe displacement displacement
resulting from resulting from vvii tt
• The displacement The displacement resulting from resulting from ½ a½ att22
Kinematic EquationsKinematic Equations
• The vector The vector representation of the representation of the position vectorposition vector
• rrff is generally not in is generally not in
the same direction asthe same direction as vvii or asor as aaii
• rrff andand vvff are generally are generally
not in the same not in the same directiondirection
Kinematic Equations, ComponentsKinematic Equations, Components
• The equations for final velocity and final The equations for final velocity and final position are vector equations, therefore position are vector equations, therefore they may also be written in component they may also be written in component formform
• This shows that two-dimensional motion at This shows that two-dimensional motion at constant acceleration is equivalent to two constant acceleration is equivalent to two independent motionsindependent motions– One motion in the One motion in the xx-direction and the other in -direction and the other in
the the yy-direction-direction
Kinematic Equations, Component Kinematic Equations, Component EquationsEquations
• vvff = = vvii + + aatt becomes becomes
vvxfxf = v = vxixi + a + axxtt
vvyfyf = v = vyiyi + a + ayytt
• rrff = r = rii + v + vii tt + ½ + ½ aatt22 becomesbecomes
xxff = x = xii + v + vxixi tt + ½ a + ½ axxtt22
yyff = y = yii + v + vyiyi tt + ½ a + ½ ayytt22
At At tt = 0 = 0, a particle moving in the , a particle moving in the xyxy plane with constant plane with constant
acceleration has a velocity of acceleration has a velocity of
and is at the origin. At and is at the origin. At tt = 3.00 s = 3.00 s, the particle's velocity is , the particle's velocity is
. Find (a) the acceleration of the . Find (a) the acceleration of the
particle and (b) its coordinates at any time particle and (b) its coordinates at any time tt..
v i 3.00̂ i 2.00̂ j m/ s
v 9.00̂ i 7.00̂ j m/ s
Projectile MotionProjectile Motion
• An object may move in both theAn object may move in both the xx and and yy directions simultaneouslydirections simultaneously
• The form of two-dimensional motion we The form of two-dimensional motion we will deal with is called will deal with is called projectile motionprojectile motion
Assumptions of Projectile MotionAssumptions of Projectile Motion
• The free-fall acceleration The free-fall acceleration gg is constant is constant over the range of motionover the range of motion
And is directed downwardAnd is directed downward
• The effect of air friction is negligibleThe effect of air friction is negligible• With these assumptions, an object in With these assumptions, an object in
projectile motion will follow a parabolic projectile motion will follow a parabolic pathpath– This path is called the This path is called the trajectorytrajectory
Verifying the Parabolic TrajectoryVerifying the Parabolic Trajectory
• Reference frame chosenReference frame chosenyy is vertical with upward positiveis vertical with upward positive
• Acceleration componentsAcceleration componentsaayy = -g = -g andand aaxx = 0 = 0
• Initial velocity componentsInitial velocity componentsvvxixi = v = vii cos cos andand vvyiyi = v = vii sin sin
II. Projectile motionII. Projectile motionMotion of a particle launched with initial velocity,Motion of a particle launched with initial velocity, v v00,, and free and free
fall accelerationfall acceleration gg..
Horizontal motion:Horizontal motion: aaxx=0=0 vvxx=v=v0x0x
tvtvxx x )cos( 0000
The horizontal and vertical motions are independent from each The horizontal and vertical motions are independent from each other.other.
II. Projectile motionII. Projectile motion
Range (R):Range (R): horizontal distance traveled by a projectile horizontal distance traveled by a projectile before returning to launch height.before returning to launch height.
II. Projectile motionII. Projectile motion
Vertical motion:Vertical motion: aayy= -g= -g
200
200 2
1)sin(
2
1gttvgttvyy y
gtvvy 00 sin )(2)sin( 02
002 yygvvy
Trajectory:Trajectory: projectile’s path.projectile’s path.
000 yx
We can findWe can find yy as a function ofas a function of xx by eliminating timeby eliminating time
tvtvx x )cos( 000 00 cosv
xt
2
000000 cos2
1
cossin
v
xg
v
xvy
200
2
0 )cos(2)(tan
v
gxxy
Horizontal range:Horizontal range: R = x-xR = x-x00; ;
Vertical displacement:Vertical displacement: y-yy-y00=0.=0.
(Maximum for a launch (Maximum for a launch angle of 45º )angle of 45º )
0000 cos)cos(
v
RttvR
022
0
2
0
2
000000
200
cos2
1tan
cos2
1
cos)sin(
2
1)sin(0
v
RgR
v
Rg
v
Rvgttv
0
202
000 2sin
cossin2 g
vv
gR
Projectile Motion – Problem Solving Projectile Motion – Problem Solving HintsHints
• Select a coordinate systemSelect a coordinate system• Resolve the initial velocity into Resolve the initial velocity into xx and and yy
componentscomponents• Analyze the horizontal motion using constant Analyze the horizontal motion using constant
velocity techniquesvelocity techniques• Analyze the vertical motion using constant Analyze the vertical motion using constant
acceleration techniquesacceleration techniques• Remember that both directions share the same Remember that both directions share the same
timetime
A rock is thrown upward from the level ground in such a way A rock is thrown upward from the level ground in such a way that the maximum height of its flight is equal to its horizontal that the maximum height of its flight is equal to its horizontal range range RR. (a) At what angle is the rock thrown? (b) Would . (a) At what angle is the rock thrown? (b) Would your answer to part (a) be different on a different planet? your answer to part (a) be different on a different planet? (c) What is the range (c) What is the range RRmaxmax the rock can attain if it is launched the rock can attain if it is launched
at the same speed but at the optimal angle for maximum at the same speed but at the optimal angle for maximum range?range?
A third baseman wishes to A third baseman wishes to throw to first base, throw to first base, 127127 feet feet distant. His best throwing distant. His best throwing speed is speed is 85 mi/h85 mi/h. (a) If he . (a) If he throws the ball horizontally throws the ball horizontally 3 ft3 ft above the ground, how far above the ground, how far from first base will it hit the from first base will it hit the ground? (b) From the same ground? (b) From the same initial height, at what upward initial height, at what upward angle must the third baseman angle must the third baseman throw the ball if the first throw the ball if the first baseman is to catch it baseman is to catch it 3 ft3 ft above the ground? (c) What above the ground? (c) What will be the time of flight in that will be the time of flight in that case?case?
x
y
v0
h=3fth=3ft
B3 B1
xmax0 xB1=38.7m
N7: N7: In Galileo’s Two New Sciences, the author states In Galileo’s Two New Sciences, the author states that “for elevations (angles of projection) which exceed or that “for elevations (angles of projection) which exceed or fall short of fall short of 4545ºº by equal amounts, the ranges are by equal amounts, the ranges are equal…” Prove this statement.equal…” Prove this statement.
xx
v0
x=R=R’?x=R=R’?
yy
θθ=45º=45º
A ball is tossed from an upper-story window of a A ball is tossed from an upper-story window of a building. The ball is given an initial velocity ofbuilding. The ball is given an initial velocity of 8.00 m/s8.00 m/s at an angle ofat an angle of 20.0°20.0° below the horizontal. It strikes the below the horizontal. It strikes the groundground 3.00 s3.00 s later. (a) How far horizontally from the later. (a) How far horizontally from the base of the building does the ball strike the ground? base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a pointHow long does it take the ball to reach a point 10.0 m10.0 m below the level of launching?below the level of launching?
A ball is tossed from an upper-story window of a A ball is tossed from an upper-story window of a building. The ball is given an initial velocity ofbuilding. The ball is given an initial velocity of 8.00 m/s8.00 m/s at an angle ofat an angle of 20.0°20.0° below the horizontal. It strikes the below the horizontal. It strikes the groundground 3.00 s3.00 s later. (a) How far horizontally from the later. (a) How far horizontally from the base of the building does the ball strike the ground? base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a pointHow long does it take the ball to reach a point 10.0 m10.0 m below the level of launching?below the level of launching?
)
Uniform Circular MotionUniform Circular Motion
Uniform circular motionUniform circular motion occurs when an occurs when an object moves in a circular path with a object moves in a circular path with a constant speedconstant speed
Uniform Circular MotionUniform Circular Motion
Uniform circular motionUniform circular motion occurs when an occurs when an object moves in a circular path with a object moves in a circular path with a constant speedconstant speed
• An acceleration exists since the An acceleration exists since the directiondirection of the motion is changing of the motion is changing – This change in velocity is related to an This change in velocity is related to an
accelerationacceleration
• The velocity vector is always tangent to The velocity vector is always tangent to the path of the objectthe path of the object
Changing Velocity in Uniform Changing Velocity in Uniform Circular MotionCircular Motion
• The change in the The change in the velocity vector is due velocity vector is due to the change in to the change in directiondirection
• The vector diagram The vector diagram showsshows v = vv = vff - v- vii
Changing Velocity in Uniform Circular MotionChanging Velocity in Uniform Circular Motion
Two triangles are similar, so Two triangles are similar, so we can write:we can write:
fifi vvvrrr
rr
vv
v
v
r
r
Dividing both parts by t and using the definitions of acceleration and velocity:
r
va
t
r
r
v
t
v 2
Centripetal AccelerationCentripetal Acceleration
• The acceleration is always perpendicular The acceleration is always perpendicular to the path of the motionto the path of the motion
• The acceleration always points toward the The acceleration always points toward the center of the circle of motioncenter of the circle of motion
• This acceleration is called theThis acceleration is called the centripetal centripetal accelerationacceleration
Centripetal AccelerationCentripetal Acceleration
• The magnitude of the centripetal The magnitude of the centripetal acceleration vector is given byacceleration vector is given by
• The direction of the centripetal The direction of the centripetal acceleration vector is always changing, to acceleration vector is always changing, to stay directed toward the center of the stay directed toward the center of the circle of motioncircle of motion
2
C
va
r
PeriodPeriod• The The periodperiod,, TT, is the time required for one , is the time required for one
complete revolutioncomplete revolution
• The speed of the particle would be the The speed of the particle would be the circumference of the circle of motion circumference of the circle of motion divided by the perioddivided by the period
• Therefore, the period is Therefore, the period is
2 rT
v
Tangential AccelerationTangential Acceleration
• The magnitude of the velocity could also The magnitude of the velocity could also be changingbe changing
• In this case, there would be aIn this case, there would be a tangential tangential accelerationacceleration
Total AccelerationTotal Acceleration
• The tangential The tangential acceleration causes acceleration causes the change in the the change in the speed of the particlespeed of the particle
• The radial The radial acceleration comes acceleration comes from a change in the from a change in the direction of the direction of the velocity vectorvelocity vector
Total Acceleration, equationsTotal Acceleration, equations
• The tangential acceleration:The tangential acceleration:
• The radial acceleration:The radial acceleration:
• The total acceleration:The total acceleration:– Magnitude Magnitude
t
da
dt
v
2
r C
va a
r
2 2r ta a a
Total Acceleration, In Terms of Unit Total Acceleration, In Terms of Unit VectorsVectors
• Define the following unit Define the following unit vectorsvectors
rr lies along the radius lies along the radius vectorvector
is tangent to the circleis tangent to the circle
• The total acceleration isThe total acceleration is
ˆˆ a n dr
2ˆ ˆt r
d v
dt r
va a a r
Uniform circular motion. SummaryUniform circular motion. Summary
Motion around a circle Motion around a circle at at constant speedconstant speed..
r
va
2
v
rT
2Period of revolution:Period of revolution:
Acceleration:Acceleration: centripetalcentripetal
Velocity:Velocity: tangent to circle in the directiontangent to circle in the direction of motion.of motion.
Magnitude of velocity and Magnitude of velocity and acceleration constant. acceleration constant. Direction varies continuously.Direction varies continuously.
vv11
vv22
x
yy
54. A cat rides a merry-go-round while A cat rides a merry-go-round while turning with uniform circular motion. At turning with uniform circular motion. At time time tt11= 2s= 2s, the cat’s velocity is: , the cat’s velocity is:
vv11= (3m/s)i+(4m/s)j= (3m/s)i+(4m/s)j, measured on an , measured on an
horizontal horizontal xy xy coordinate system. At time coordinate system. At time t=5st=5s its velocity is: its velocity is: vv22= (-3m/s)i+(-4m/s)j= (-3m/s)i+(-4m/s)j. .
What is (a) the magnitude of the cat’s What is (a) the magnitude of the cat’s centripetal acceleration?centripetal acceleration?
Figure represents the total acceleration of a particle Figure represents the total acceleration of a particle moving clockwise in a circle of radius moving clockwise in a circle of radius 2.50 m2.50 m at a certain of at a certain of time. At this instant, find (a) the radial acceleration, (b) the time. At this instant, find (a) the radial acceleration, (b) the speed of the particle, and (c) its tangential acceleration.speed of the particle, and (c) its tangential acceleration.
A ball swings in a vertical circle at the end of a ropeA ball swings in a vertical circle at the end of a rope 1.50 m1.50 m long. When the ball islong. When the ball is 36.936.9 past the lowest point on its way up, past the lowest point on its way up, its total acceleration is . At that instant, its total acceleration is . At that instant, (a) sketch a vector diagram showing the components of its (a) sketch a vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the acceleration, and (c) determine the speed and velocity of the ball.ball.
2 2 .5 ˆ i 2 0 .2 ˆ j m / s 2
Relative VelocityRelative Velocity• Two observers moving relative to each other generally Two observers moving relative to each other generally
do not agree on the outcome of an experimentdo not agree on the outcome of an experiment• For example, observers For example, observers AA and and B B below see different below see different
paths for the ballpaths for the ball
Galilean RelativityGalilean Relativity
The observer in the The observer in the truck throws a ball truck throws a ball straight upstraight up– It appears to move in It appears to move in
a vertical patha vertical path– The law of gravity The law of gravity
and equations of and equations of motion under uniform motion under uniform acceleration are acceleration are obeyedobeyed
Galilean RelativityGalilean Relativity
There is a stationary observer on the groundThere is a stationary observer on the ground– Views the path of the ball thrown to be a parabolaViews the path of the ball thrown to be a parabola– The ball has a velocity to the right equal to the The ball has a velocity to the right equal to the
velocity of the truckvelocity of the truck
Galilean Relativity – conclusionGalilean Relativity – conclusion
The two observers disagree on the shape of the ball’s path
Both agree that the motion obeys the law of gravity and Newton’s laws of motion
Both agree on how long the ball was in the air
Conclusion: There is no preferred frame of reference for describing the laws of mechanics
Relative VelocityRelative Velocity
• Reference frameReference frame SS is is stationarystationary
• Reference frameReference frame SS’’ is is moving atmoving at vvoo
– This also means thatThis also means that SS moves atmoves at –v–voo relative relative toto SS’’
• Define timeDefine time tt = 0 = 0 as as that time when the that time when the origins coincideorigins coincide
The coordinates of some event in frame S are (The coordinates of some event in frame S are (xx,,yy,,zz,,tt). ). Now what are the coordinates of the event (Now what are the coordinates of the event (xx,,yy,,zz,,tt) in ) in SS''?? It's easy to seeIt's easy to see t't' = = tt - we synchronized the clocks when - we synchronized the clocks when O‘O‘passed passed OO. Also, evidently, . Also, evidently, y'y' = = yy and and z'z' = = zz, from the figure., from the figure.We can also see that We can also see that x x = = x'x' + +vtvt. Thus (. Thus (x,y,z,tx,y,z,t) in ) in SScorresponds to (corresponds to (x',y',z', t'x',y',z', t' ) in ) in S'S', where , where
That's how That's how positionspositions transform - these are known as the transform - these are known as the GalileanGalilean transformations transformations..
tt
zz
yy
vtxx
What about What about velocitiesvelocities ? The velocity in ? The velocity in S'S' in the in the x'x' direction direction
This is just the addition of velocities formulaThis is just the addition of velocities formula
vuvd t
d xv tx
d t
d
d t
xd
td
xdu xx
)(
vuu xx
How does How does accelerationacceleration transform? transform?
dt
duvu
dt
d
dt
ud
td
ud xx
xx
)(
the the acceleration is the sameacceleration is the same in both frames. This in both frames. This again is obvious - the acceleration is the rate of again is obvious - the acceleration is the rate of change of velocity, and the velocities of the same change of velocity, and the velocities of the same particle measured in the two frames differ by a particle measured in the two frames differ by a constantconstant factor - the relative velocity of the two factor - the relative velocity of the two frames.frames.
xx aa
SinceSince vv is constant we haveis constant we have
Relative motionRelative motion
1D
Particle’s velocity depends on reference frameParticle’s velocity depends on reference frame
BAPBPA vvv Frame moves at constant velocityFrame moves at constant velocity
PBPABAPBPA aavdt
dv
dt
dv
dt
d )()()(
Observers on different frames of reference measure the same accelerationObservers on different frames of reference measure the same accelerationfor a moving particle if their relative velocity is constant.for a moving particle if their relative velocity is constant.
BAPBPA XXX
dt
dX
dt
dX
dt
dX BAPBPA
75.75. A sled moves in the negative x direction at speedA sled moves in the negative x direction at speed vvss while while
a ball of ice is shot from the sled with a velocitya ball of ice is shot from the sled with a velocity vv00= v= v0x0xi+ vi+ v0y0yjj
relative to the sled. When the ball lands, its horizontal relative to the sled. When the ball lands, its horizontal displacementdisplacement ΔΔxxbgbg relative to the ground (from its launch relative to the ground (from its launch
position to its landing position) is measured. The figure givesposition to its landing position) is measured. The figure gives ΔΔxxbgbg as a function ofas a function of vvss.. Assume it lands at approximately its Assume it lands at approximately its
launch height. What are the values of (a)launch height. What are the values of (a) vv0x0x and (b)and (b) vv0y0y? The ? The
ball’s displacementball’s displacement ΔΔxxbsbs relative to the sled can also be relative to the sled can also be
measured. Assume that the sled’s velocity is not changed measured. Assume that the sled’s velocity is not changed when the ball is shot. What iswhen the ball is shot. What is ΔΔxxbsbs whenwhen vvss is (c)is (c) 5m/s5m/s and (d)and (d)
15m/s15m/s??
120.120. A hang glider isA hang glider is 7.5 m7.5 m above ground level with a velocity ofabove ground level with a velocity of 8m/s8m/s at at an angle ofan angle of 3030ºº below the horizontal and a constant acceleration ofbelow the horizontal and a constant acceleration of 1m/s1m/s22, , up. (a) Assumeup. (a) Assume t=0t=0 at the instant just described and write an equation for at the instant just described and write an equation for
the elevationthe elevation yy of the hangof the hang glider as a function ofglider as a function of tt, with, with y=0y=0 at ground at ground level. (b) Use the equation to level. (b) Use the equation to determine the value ofdetermine the value of t t whenwhen y=0y=0. . (c) Explain why there are two (c) Explain why there are two solutions to part (b). Which one solutions to part (b). Which one represents the time it takes the represents the time it takes the hang glider to reach ground level? hang glider to reach ground level? (d) how far does the hang glider (d) how far does the hang glider travel horizontally during the interval betweentravel horizontally during the interval between t=0t=0 and the time it reaches and the time it reaches the ground? For the same initial position and velocity, what constant the ground? For the same initial position and velocity, what constant acceleration will cause the hang glider to reach ground level with zero acceleration will cause the hang glider to reach ground level with zero velocity? Express your answer in terms of unit vectors.velocity? Express your answer in terms of unit vectors.
xx
yy
v0= 8m/sh=7.5mh=7.5m
00
30º
40. 40. A ball is to be shot from level ground with certain speed. A ball is to be shot from level ground with certain speed. The figure below shows the rangeThe figure below shows the range RR it will have versus the it will have versus the launch anglelaunch angle θθ00 at which it can be launched. The choice ofat which it can be launched. The choice of θθ00
determines the flight time; letdetermines the flight time; let ttmaxmax represent the maximum represent the maximum
flight time. What is the least speed the ball will have during its flight time. What is the least speed the ball will have during its flight ifflight if θθ00 is chosen such as that the flight time isis chosen such as that the flight time is 0.5t0.5tmaxmax??
R(m)R(m)
θθ00
100100
200200240240
