Physics and Astronomy Outreach Program at the University of British Columbia

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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?

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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.

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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 :

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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 :

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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

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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