A monkey is sitting in a tree For some reason you want to shoot it But it is an astute cheeky even monkey and will let go of the branch he is swinging from at the same instant you shoot your gun or arrow or blowpipe or whatever weapon you use ID: 679427
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Slide1
Shoot the Monkey!Slide2
Shoot the Monkey!
A monkey is sitting in a tree. For some reason you want to shoot it. But it is an astute (cheeky, even) monkey and will let go of the branch he is swinging from at the same instant you shoot your gun or arrow or blowpipe or whatever weapon you use.
So, where should you aim to guarantee hitting the blighter?
Ignoring air resistance, etc.Slide3
Will the hunter hit his prey?Slide4
Monkey’s initial position
Monkey’s position after time
seconds, falling under gravity,
m/s
2
Path of projectileSlide5
Projectile
Monkey
After time,
Vertical displacement:
Horizontal displacement:
Time for projectile to reach fall-line of monkey
Vertical displacement
at
:
Projectile
Monkey
Vertical displacement:
Horizontal displacement:
Time for projectile to reach fall-line of monkey
Equating vertical displacements
gives:
Re-arranging gives:
So, just aim at the monkey!
Slide6
Projectile
Monkey
After time,
Vertical displacement:
Horizontal displacement:
Time for projectile to reach fall-line of monkey
Vertical displacement
at
:
Projectile
Monkey
Vertical displacement:
Horizontal displacement:
Time for projectile to reach fall-line of monkey
Equating vertical displacements
gives:
Re-arranging gives:
So, just aim at the monkey!
Slide7
Monkey’s initial position
Another way of looking at it is to consider the position vectors of each object and note how they change with time.
Considering the monkey’s position vector first:
Slide8
Monkey’s initial position
Path of projectile
Another way of looking at it is to consider the position vectors of each object and note how they change with time.
Now considering the projectile’s position vector:
Slide9
Monkey’s initial position
Another way of looking at it is to consider the position vectors of each object and note how they change with time.
Considering both position vectors:
so collision hereSlide10
Algebraicly:
For a collision
So
Equating horizontally:
So
Equating vertically
:
So
Slide11
Will the hunter hit his prey?Slide12
HA, YOU MISSED ME!
Will the hunter hit his prey?Slide13
Will the hunter hit his prey?Slide14
Will the hunter hit his prey?
AAAARGHHHH!Slide15
Will the hunter hit his prey?
First scenario – with line of fireSlide16
Will the hunter hit his prey?
Second scenario – with line of fireSlide17
This scenario hits the target
(line of fire is on the target)
This scenario does not hit the target
(line of fire is not on the target)Slide18
No matter what initial velocity the projectile has the only criteria for hitting the target is that the projectile is aimed directly at it.
The initial velocity only determines
when
the target is hit. A small velocity will obviously take longer and so hunter and target should be on opposite sides of a very deep canyon!Slide19
Another way of thinking about it is to note that each object (projectile and monkey) have identical vertical acceleration – gravity. So, subtract that motion from each object and analyse. It is now as though gravity is zero so the answer of aiming directly at the target becomes obvious.
Each object has “fallen” the same distance.Slide20Slide21
RESOURCESSlide22
Will the hunter hit his prey?Slide23Slide24
Shoot the Monkey!
26.5 m
33.92 m
21.5 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide25
Shoot the Monkey!
27.5 m
35.2 m
23.5 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide26
Shoot the Monkey!
28.5 m
36.48 m
25 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide27
Shoot the Monkey!
29.5 m
37.76 m
26.2 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide28
Shoot the Monkey!
32.5 m
41.6 m
26.2 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide29
Shoot the Monkey!
37.5 m
48.0 m
30 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide30
Shoot the Monkey!
38.5 m
49.2 m
31.2 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide31
Shoot the Monkey!
39.5 m
50.56 m
32.2 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide32
Shoot the Monkey!
40.5 m
51.84 m
33.4 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide33
Shoot the Monkey!
41.5 m
53.12 m
35.4 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide34
Shoot the Monkey!
42.5 m
54.4 m
36 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26Slide35
Shoot the Monkey!
43.5 m
55.68 m
39.7 ms
-1
The instant you shoot, the monkey lets go and falls to the ground.
What angle ensures you hit him?
9.81 ms
-2
(Not to scale)
SIC_26