Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia
Lecture Notes
Animals
Cheetah Chase
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• Physics: To solve a problem involving constant acceleration and constant motion.
• Biology: Learn about speed and acceleration characteristics of cheetahs and Thomson’s gazelles.
Falling CatsGoals
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• Physics: During a chase animals go through an initial acceleration phase and a phase at which they can travel at top speed.
• Biology: Cheetahs can maintain top speed for a limited distance.
Cheetah ChaseBig Ideas
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Gazelle vs Cheetah
[4] [2]
Cheetah[1]
• Max speed: 110 km/h (30.6 m/s)
• Acceleration 0 – 96 km/h in 3.0 s! (8.9 m/s2)
• Catching prey should be easy, right?
Gazelle[2,3]
• Max speed: 70 km/h (19.4 m/s)
• Acceleration: ag ~ 4.5 m/s2
• Can make sharp turns
• Good endurance
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Cheetah ChaseQuestion
Cheetahs can only keep up their maximumspeed for ~ 400 m, so they need to be relatively close to their prey [5].
What is the maximum distance away froma gazelle for a cheetah to have a chance of catching the gazelle?
Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia
Cheetah Chase
Assumptions
• The cheetah and gazelle start accelerating at the same time.
• The rate of acceleration from rest to top speed is constant.
• The gazelle only runs straight.
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Cheetah ChaseVisualization
dc1 dc2 = 400 m
dg2
A B C D
• Cheetah starts at A, accelerates until B and continues at top speed until E.• Gazelle starts at C, accelerates until D and continues at top speed until E.
E
• Answer is the distance from A to C.
dg1
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Cheetah ChaseStrategy
• Calculate the distance travelled and the time taken for the cheetah to initially accelerate and then travel 400 m.
• Calculate the distance the gazelle travelled during this time.
• Find the difference between how far the gazelle and the cheetah travelled.
Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia
Cheetah ChaseCalculations
Cheetah accelerates for time tc1
s4.3sm
sm9.8
)06.30(2-
1-
1
c
icfcc a
vvt
Distance travelled in time tc1
m5.52)sm9.8)(2(
)sm6.30(2 2
212
1
c
fcc a
vd
Cheetah acceleration phase :
Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia
Cheetah ChaseCalculations
Cheetah travels at top speed for time tc2
s1.13sm6.30
m4001
22
fc
cc v
dt
Total time taken from start to finish
tc tc1 tc2 3.4 s13.1 s 16.5 s
Cheetah constant velocity phase :
Total distance cheetah travelled
m 452.5m)5.52400(21 ccc ddd
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Cheetah ChaseCalculations
Gazelle accelerates for time tg1
s3.4sm
sm5.4
)04.19(2-
1-
1
g
igfgg a
vvt
Distance travelled in time tg1
m0.42)sm5.4)(2(
)sm4.19(2 2
212
1
g
fgg a
vd
Gazelle acceleration phase :
Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia
Cheetah ChaseCalculations
Gazelle travels at top speed for time tg2
tg2 tc tg1 (16.5 4.3) s 12.2 s
Gazelle constant velocity phase :
Total distance gazelle travelled
m 278.7 m)7.2360.42(21 ggg ddd
Gazelle travels distance dg2 in time tg2
m7.23622 ggfg tvd
Physics and Astronomy Outreach Program at the University of British ColumbiaPhysics and Astronomy Outreach Program at the University of British Columbia
Cheetah ChaseCalculationsDistance gazelle ahead of cheetah :
dc1 dc2 = 400 m
dg2
A B C D E
dg1
m 278.7 21 ggg ddd
m 452.5 21 ccc ddd
m 174 m 278.7) - (452.5 gc dd
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Cheetah ChaseConclusions
The gazelle could not possibly be caughtby the cheetah if the gazelle is more than174 m ahead of the cheetah (assuming they both start running at the same time.).
This assumes linear motion. In real life gazelles use turning maneuvers to escape and the cheetah will quietly come within 10-30 m and then initiate a chase [4].
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Cheetah ChaseBibliography
[1] http://openlearn.open.ac.uk/mod/resource/view.php?id=344455 [2] http://en.wikipedia.org/wiki/Thomson's_Gazelle [3] McNeill Alexander, “Principles of animal locomotion”, Princeton University Press, p. 3, (2003)[4] http://en.wikipedia.org/wiki/Cheetah [5] http://www.lioncrusher.lunarpages.com/animal.asp?animal=44
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