High-speed motion pictures (3500 frames second) of a jumping, 210 g flea yielded

Question

High-speed motion pictures (3500 frames second) of a jumping, 210 g flea yielded the data used to plot the graph given in the figure below (See "The Flying Leap of the Flea" by M Rothschild, Y. Schlein, K. Parker, C. Neville, and s. Sternberg in the November 1973 Scientific American.) This flea was about 2 mm long and jumped at a nearly vertical take-off angle. Use the graph to answer the questions (Hint model the graph as two straight lines) 150 F 100E 50 0.5 1.0 1.5 2.0 2.5 Time (in milliseconds) (a) Is the acceleration of the flea ever zero? If so, when is it first zero? Justify your answer (Enter 0 if the acceleration of the flea is never zero.) (b) Find the maximum height of the flea reached in the first 2.5 ms. (c) Find the flea's acceleration at the given times below (d) Find the flea's height at the given times below t 0.5 ms t=1.0 ms t= 1.5ms

Explanation / Answer

a) yes acceletaion of the flea will be 0 from the time when graph becomes parallel to time axis

graph parallel to time axis denotes there is no change in speed oder time and since

acceleration = change in speed / time

and change in speed = 0 so,

acceleration will be 0

acceleration will be first 0 when graph starts to become parallel to time axis which is 1.25 ms

b) if we model the graph as two straight line we can see that speed ad 0.5 milli sec will be 50 cm/s or 0.5 m/s

from this we can take out the acceleration as

acceleration = (0.5 - 0) / (0.5 * 10^-3 - 0)

acceleration = 1000 m/s^2

height = 0.5 * acceleration * time^2

height at 1.25 ms = 0.5 * 1000 * (1.25 * 10^-3)^2

height at 1.25 ms = 0.00078125 m

after 1.25 ms acceleration cease to exist and height will be

height = speed * time

after 1.25 ms speed will become constant at 125 cm/s or 1.25 m/s so,

height = 1.25 * (2.5 * 10^-3 - 1.25 * 10^-3)

height = 0.0015625 m

maximum height at 2.5 ms = 0.00078125 + 0.0015625

maximum height at 2.5 ms = 0.00234375 m or 0.234375 cm

c) from graph we can see that slope of the graph will give us the acceleration
so,
acceleration at 0.5 ms = 1000 m/s^2

acceleration at 1 ms = 1000 m/s^2

acceleration at 1.5 sec = 0 m/s^2

d) height at 0.5 ms = 0.5 * 1000 * (0.5 * 10^-3)^2

height at 0.5 ms = 0.000125 m or 0.0125 cm

height at 1 ms = 0.5 * 1000 * (1 * 10^-3)^2

height at 1 ms = 0.0005 m or 0.05 cm

height at 1.5 ms = 0.00078125 + 1.25 * (1.5 * 10^-3 - 1.25 * 10^-3)

height at 1.5 ms = 0.00109375 m or 0.109375 cm

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