Lesson Thread I Full Projectile Motion
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Transcript of Lesson Thread I Full Projectile Motion
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Intellectual Property of Mr. Gary 1
Lesson ThreadI
Full Projectile Motion
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Intellectual Property of Mr. Gary 2
Projectile Motion
Lesson I
Determine the Maximum Height
Part a
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Intellectual Property of Mr. Gary 3
A batter on the ladies softball team hits a fly ball with an initial velocity of 35 m/s at an angle of 27° above the horizontal. Calculate (a) the maximum height the ball reaches above the ground, (b) the x, y – components of the velocity of the ball at the maximum height, (c) the time it takes the ball to reach the maximum height, (d) the maximum range the ball traveled from the batter.
Question: Part a
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Intellectual Property of Mr. Gary 4
0v
0 0 0 cosxv v
0 0 0sinyv v
27
1 1 1 cosxv v 1 1 1sinyv v
27
2 2 2cosxv v
2 2 2sinyv v
0v
y��������������
Sketch: Part a
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Intellectual Property of Mr. Gary 5
212
2 20
sin sin
sin sin
( sin ) ( sin ) 2
gv v t
y v t g t
v v g y
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������������������������������������������
�������������������������� ��
0
212
2 2
cos cos
cos
( cos ) ( cos ) 2
v v a t
x v t a t
v v a x
Component Equations: Part a
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Intellectual Property of Mr. Gary 6
max max max
max
ma
20
x
20
Eq. 2.11c
Find the maximum height
( sin ) ( sin ) 2
The maximum vertical height has an angle 0
( sin(0 )
f f
f
f
v v g y
RPC
v
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max
max
m
0 2 20
ax
0
20 0
2 20 00 0
) ( sin ) 2
0 ( sin ) 2
Perform all necesary algebraic steps to solve for y .
( sin ) 0 [ ( si )
n
v g y
v g y
v v
�������������������������� ��
�������������������������� ��
2
0
0
0( sin )v 0
20
max
ma0
20
x
0
] 2
( sin ) 0 2
2( sin )
2
g y
v g y
v
g
����������������������������
�������������������������� ��
��������������
1g�������������� 1
max
2
y��������������
1g��������������
20 0
1
2
max
max
0 0
( sin )
2
( sin )
2
vy
g
vy
g
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��������������
��������������
��������������20
(35 sin)(27 ))
2(
m s
209.8 )
12.88mm s
Solution: Part a
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Intellectual Property of Mr. Gary 7
Projectile Motion
Lesson IIDetermine the x, y – components at the
maximum height.Part b
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Intellectual Property of Mr. Gary 8
A batter on the ladies softball team hits a fly ball with an initial velocity of 35 m/s at an angle of 27° above the horizontal. Calculate (a) the maximum height the ball reaches above the ground, (b) the x, y – components of the velocity of the ball at the maximum height, (c) the time it takes the ball to reach the maximum height, (d) the maximum range the ball traveled from the batter.
Question: Part b
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Intellectual Property of Mr. Gary 9
0v
0 0 0 cosxv v
0 0 0sinyv v
27
1 1 1 cosxv v 1 1 1sinyv v
27
2 2 2cosxv v
2 2 2sinyv v
0v
y��������������
Sketch: Part b
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Intellectual Property of Mr. Gary 10
212
2 20
sin sin
sin sin
( sin ) ( sin ) 2
gv v t
y v t g t
v v g y
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������������������������������������������
�������������������������� ��
0
212
2 2
cos cos
cos
( cos ) ( cos ) 2
v v a t
x v t a t
v v a x
Component Equations: Part b
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Intellectual Property of Mr. Gary 11
1 1 1
1
Solving for the y-component
sin
(35 ) s 0in( )
y
y
v v
RPC
v m s
0
1 0y m sv
1 1 1
1
Solving for the x-component
cos
(35 ) c 0os( )
x
x
v v
RPC
v m s
1
1 35x mv s
Solution: Part b
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Intellectual Property of Mr. Gary 12
Projectile Motion
Lesson IIICalculate the time taken to reach the
maximum heightPart c
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Intellectual Property of Mr. Gary 13
A batter on the ladies softball team hits a fly ball with an initial velocity of 35 m/s at an angle of 27° above the horizontal. Calculate (a) the maximum height the ball reaches above the ground, (b) the x, y – components of the velocity of the ball at the maximum height, (c) the time it takes the ball to reach the maximum height, (d) the maximum range the ball traveled from the batter.
Question: Part c
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Intellectual Property of Mr. Gary 14
0v
0 0 0 cosxv v
0 0 0sinyv v
27
1 1 1 cosxv v 1 1 1sinyv v
27
2 2 2cosxv v
2 2 2sinyv v
0v
y��������������
Sketch: Part c
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Intellectual Property of Mr. Gary 15
212
2 20
sin sin
sin sin
( sin ) ( sin ) 2
gv v t
y v t g t
v v g y
������������� �
������������������������������������������
�������������������������� ��
0
212
2 2
cos cos
cos
( cos ) ( cos ) 2
v v a t
x v t a t
v v a x
Component Equations: Part c
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Intellectual Property of Mr. Gary 16
Solution: Part c
1 01 0
1
Choose Eq. 2.11a: y-component
sin sin
sin 0( )
y yy y
y
gv v t
RPC
v
������������� �
00 0
0 00 0
sin
( sin ) 0 [ ( sin )
y y
y yy y
gv t
v v
������������� �
0
0 0( sin )y yv 0
0 0
0 0
]
( sin )
( sin )
y y
y y
g
g
gt
t
v
tv
��������������
��������������
��������������
1(35 )sin(27 )
(
m s
219.8 )
1.62
m s
t s
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Intellectual Property of Mr. Gary 17
Projectile Motion
Lesson IVCalculate the Range the particle travels
horizontallyPart d
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Intellectual Property of Mr. Gary 18
A batter on the ladies softball team hits a fly ball with an initial velocity of 35 m/s at an angle of 27° above the horizontal. Calculate (a) the maximum height the ball reaches above the ground, (b) the x, y – components of the velocity of the ball at the maximum height, (c) the time it takes the ball to reach the maximum height, (d) the maximum range the ball traveled from the batter.
Question: Part d
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Intellectual Property of Mr. Gary 19
0v
0 0 0 cosxv v
0 0 0sinyv v
27
1 1 1 cosxv v 1 1 1sinyv v
27
2 2 2cosxv v
2 2 2sinyv v
0v
y��������������
Sketch: Part d
x = Range
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Intellectual Property of Mr. Gary 20
212
2 20
sin sin
sin sin
( sin ) ( sin ) 2
gv v t
y v t g t
v v g y
������������� �
������������������������������������������
�������������������������� ��
0
212
2 2
cos cos
cos
( cos ) ( cos ) 2
v v a t
x v t a t
v v a x
Component Equations: Part d
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Intellectual Property of Mr. Gary 21
212
210
0
20
0
Choose Eq. 2.11b
cos
There is no acceleration in x-direction
cos
x v t a t
x v t a t
0 0
0
0
0
cos 0
cos
We can now double the time calculated in Part b and substitute
that value into .
(35 )cos(27 ) (1.62s)
(35
t
2
x v t
RPC
x v t
x m s
x m s
1)cos(27 ) (3.24 s
1
101. 4
)
0x m
Solution: Part d
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