Class04_AP1_Physics_Lesson (1)doeiwjoejfo.pdf

Physics

1. Class 4: Projectile Motion and Relative Motion
AP Physics 1
Dr. Timothy Leung
Winter 2025
Meritus Academy
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2. Projectile Motion

3. Projectile Motion
Aprojectileis an object that is launched with an initial velocity of⃗V0along a parabolic
trajectory and accelerates only due to gravity.
xy⃗V0⃗v0⃗u0θ
•x-axis:horizontal, pointingforward
•y-axis:vertical, pointingup
•Angleθmeasuredabovethe horizontal, i.e.
•θ>0 if above horizontal
•θ<0 if below horizontal
•The origin is located at where the projectile is launched
•Consistent with the right-handed Cartesian coordinate
system
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4. Horizontal Direction
The initial velocity⃗V0can be decomposed into itsxandycomponents using the launch
angleθ:
⃗V0=⃗u0+⃗v0=[V0cosθ]ˆx+[V0sinθ]ˆy
There is no horizontal acceleration (i.e.ax=0), therefore thex-component of velocity
u(t) =u0is constant. The kinematic equations reduce to a single equation:
x(t) =u0t=[V0cosθ]t
wherexis the horizontal position at timet
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5. Vertical Direction
There is a constant vertical acceleration due to gravity alone, i.e.ay=−g.
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The
important equation is this one:
y(t) =v0t−
1
2
gt
2
=[V0sinθ]t−
1
2
gt
2
These two kinematic equations describing they-component of velocityvy(t)may also
be useful:
vy=[V0sinθ]−gt
v
2
y=
Θ
V
2
0sin
2
θ
Λ
−2gy
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ayisnegativedue to the way we defined the coordinate system: the+ydirection isup. Acceleration is
down, therefore negative.
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6. Solving Projectile Motion Problems
Horizontal (x) and vertical (y) positions are independent of each other, but there are
variables shared in both directions:
•Timet
•Launch angleθ(above the horizontal)
•Initial speedV0
When solving any projectile motion problems
•Twoequations withtwounknowns
•If an object lands on an incline, there will be a third equation relatingxandy
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7. Projectile Motion Example
Example: While hiking in the wilderness, you come to
a cliff overlooking a river. A topographical map shows
that the cliff is 291 m high and the river is 68.5 m wide
at that point. You throw a rock directly forward from
the top of the cliff, giving the rock a horizontal velocity
of 12.8 m/s.
(a)
(b)
bottom of the cliff?
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8. Projectile Motion Example
Example: A golfer hits the golf ball off the tee, giving it an initial velocity of 32.6 m/s at
an angle of 65
◦
with the horizontal. The green where the golf ball lands is 6.30 m higher
than the tee, as shown below.
(a)
(b) horizontally?
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9. Projectile Motion Example
Example: You are playing tennis with a friend on tennis courts surrounded by a 4.8 m
fence. Your friend hits the ball over the fence, and you offer to retrieve it. You find the
ball at a distance of 12.4 m on the other side of the fence. You throw the ball at an angle
of 55.0
◦
with the horizontal, giving it an initial velocity of 12.1 m/s. The ball is 1.05 m
above the ground when you release it. Does the ball go over the fence, hit the fence, or
hit the ground before reaching the fence?
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10. Symmetric Trajectory
A projectile’s trajectory issymmetricif the object lands at the same height as when it
launched. The angleθis measured above the horizontal.
Time of flight:
T=
2v0sinθ
g
Horizontal range:
R=
v
2
0
sin(2θ)
g
Maximum height:
H=
v
2
0
sin
2
θ
2g
These equations are not given in the equation sheet for your AP exam, but they are
relatively easy to derive.
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11. Maximum Range
R=
v
2
0
sin(2θ)
g
•Maximum range occurs atθ=45
◦
•For a given initial speedv0and rangeR, launch angleθis given by:
θ1=
1
2
sin
−1
`
Rg
v
2
0
´
But there is another angle thatgives the same range!
θ2=90
◦
−θ1
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12. Relative Motion

13. Relative Motion
All motion quantities must be measured relative to a frame of reference•Frame of reference: a coordinate system from which physical measurements are
made
•There is no absolute motion/rest: all motions are relative
•Principle of relativity: All laws of physics are equal in all inertial (non-accelerating)
frames
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14. Relative Motion
x
′
Cy
′
xByA
The position and motion ofAcan be calculated from two
frames of reference (coordinate systems)
•Bwith axesx,y
•Cwith axesx
′
,y
′
The two reference frames may (or may not) be moving
relative to each other. The motion of the two reference
frames affect how motion ofAis calculated.
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15. Relative Motion
x
′
Cy
′
xByA⃗rAC⃗rAB
The position ofA(t)can be described by
•⃗rAB(t)(relative toB)
•⃗rAC(t)(relative toC)
Without needing mathematically rigorous vector
notation, it is obvious that⃗rAB(t)and⃗rAC(t)are
different vectors
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16. Relative Motion
x
′
Cy
′
xBy⃗rBCA⃗rAC⃗rAB
The position vector of the origins of the two
reference frames is given by⃗rBC(t)
•The vector pointing from the origin of frameC
to the origin of frameB
•If the two frames are moving relative to each
other, then⃗rBCis also a function of time
Without using vector notations, the relationship
between the vectors is obvious:
⃗rAC(t)⃗rAB(t)⃗rBC(t)
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17. Relative Motion
x
′
Cy
′
xByA⃗rAC⃗rAB⃗rBC
Starting from the definition ofrelative position:
⃗rAC=⃗rAB+⃗rBC
Dividing by time, we get a similar equation for
relative velocity:
⃗vAC=⃗vAB+⃗vBC
andrelative acceleration:
⃗aAC=⃗aAB+⃗aBC
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18. Relative Velocity
In classical mechanics, the equation for relative velocities follows theGalilean velocity
addition rule, which applies to speeds much less than the speed of light:
⃗vAC=⃗vAB+⃗vBC
The velocity ofArelative to reference frameCis the velocity ofArelative to reference
frameB, plus the velocity ofBrelative toC. If we add another reference frameD, the
equation becomes:
⃗vAD=⃗vAB+⃗vBC+⃗vCD
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19. Relative Velocity Example
If an airplane (P) flies in windy air (A) we must consider the velocity of the airplane
relative to air, i.e.⃗vPAand the velocity of the air relative to the groundG, i.e.⃗vAG. The
velocity of the airplane relative to the ground is therefore
⃗vPG=⃗vPA+⃗vAG
Simple example:If an airplane is flying at a constant velocity of 253 km/h south relative
to the air and the air velocity is 24 km/h east, what is the velocity of the airplane
relative to Earth?
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20. Example
Example: You are the pilot of a small plane and want to reach an airport, 600 km due
south, in 4.0 h. A wind is blowing at 50 km/h [S 35
◦
E]
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. With what heading and
airspeed should you fly to reach the airport on time?
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This is not a notation style that we use in AP Physics, but we should nevertheless be familiar with the
concepts presented in this example
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