You\'re on an alien planet, and you drop a ball from a certain height h above th

Question

You're on an alien planet, and you drop a ball from a certain height h above the ground. You measure that it takes a time t for the ball to hit the ground from the moment it's dropped. a. Suppose you perform this experiment at two planets A add B using the same drop height h. On planet B you observe a longer drop time - does this indicate that the gravity on that planet is stronger or weaker than on planet A? Explain. b. Suppose you perform this experiment on two other planets C and D, and you find that you can make the drop times to be equal on both planets provided you drop the ball from higher on planet C. Which planet do you think has stronger gravity? Explain. c. Assuming that, the acceleration due to gravity near the surface of a planet, is proportional to for some exponents a and b. Use dimensional analysis to determine these exponents? d. Are your mathematical results for the exponents a and b in part c. consistent with your assertions in parts a. and b.? Explain.

Explanation / Answer

a)

consider the free fall motion of the drop

h = (0.5) a t2

a = 2h/t2

so acceleration due to gravity is inversly related to square of the drop time .

hence , longer the time , smaller will be the value of acceleration due to gravity

since drop time is longer on planet B , the acceleration due to gravity is weaker on planet B

b)

given that :

hc > hd

(0.5) ac tc2 > (0.5) ad td2

but tc = td

hence ac > ad

planet c has stronger gravity

c)

gplanet = ha tb

replacing each by dimensional formula

[M0 L1 T-2] = [M0 L1 T0]a [M0 L0 T1]b

[M0 L1 T-2] = [M0 La T0] [M0 L0 Tb]

[M0 L1 T-2] = [M0 La Tb]

comparing the exponents both side

a = 1 , b = -2

d)

yes they are consistent

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