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Assume that Galileo had attempted to drop two objects off of the Tower of Pisa. (a) If he had released the objects from his hands, but dropped one slightly sooner than the second with a time difference of ∆t = 0.1 s, then what would be the vertical separation of the two objects just before hitting the ground? (b) What release accuracy ∆t would he need so that the two objects would have a vertical separation of less than 1 cm just prior to hitting the ground? (ignore any effects of air friction.)

Assume  that Galileo  had  attempted  to drop  two objects  off of the Tower  of Pisa.  (a) If he had released  the objects  from  his hands,  but dropped  one slightly  sooner  than the second  with a time difference  of  ∆t = 0.1 s, then what would  be the vertical separation   of  the  two  objects  just  before  hitting   the  ground? (b)  What  release   accuracy ∆t would he need so that the two objects  would  have a vertical  separation   of less than  1 cm just prior to hitting the ground? (ignore any effects of air friction.)

Grade:10

1 Answers

Jitender Pal
askIITians Faculty 365 Points
6 years ago
(a) The distance say (y1) travelled by the first object in time t1 using equation (1) as:
235-794_10.PNG
Here g is the acceleration due to gravity 9.81 m/s2.
Since the object is about to hit the ground, the distance y1 is equal to height of tower say (h). Substitute the initial speed v0y = 0 m/s and the value of y1 to obtain time t as:
235-498_11.PNG
Therefore the time taken by the object to reach the ground is \sqrt{2h/g}
Since the second object was released 0.1 seconds later, the time say t2 for which the object has traveled relative to the first one is .t – 0.1 The distance say y2 travelled by the second object in time t2 is given as:
235-95_12.PNG
Here is the acceleration due to gravity 9.81 m/s2.
235-2379_13.PNG
235-413_14.PNG
235-235_15.PNG

It is clear from above, that for the value of h in meters, the term m1/2 cancels out, thereby bringing the units of time \Delta tas seconds.
Therefore the time difference that would account for separation of 1 cm is235-615_16.PNG

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