Question

A penny, which has a mass 2.5 grams and is made of zinc, is released from rest just beneath the surface of some 80 proof vodka. (80 proof means 40% ethyl alcohol and 60% water, by volume). If the height of the vodka in the bottle is 25cm, how long will it take for the penny to reach the bottom? (Ignore drag and other frictional effects. Assume the only effect of the fluid is buoyancy.)

Answer 1

weight of the penny = mg

=2.5*10^-3*9.81

=0.0245 N

volume of the penny= mass/density

density of zinc = 7140 kg/m^3

so volume of the penny = 2.5*10^-3/7140

=3.5*10^-7 m^3

in the alcohol section,

balancing the forces,

Weight - Buoyant force = ma

or 0.0245 - rho*V*g = ma

or 0.0245 - 789*3.5*10^-7*9.81 = 2.5*10^-3*a

or a=8.716 m/s^2

so time taken to cover 0.25m be t.so,

0.25 = 0 + 0.5*at^2

or 0.25 = 0.5*8.716*t^2

or t=0.239 s

since the length of the alcohol section is 25 cm, the length of the water section = 0.6*25/0.4

=37.5 cm

velocity of the penny after it passes through the alcohol section = at

=8.716*0.239

=2.08 m/s

now for the water part,

balancing all the forces,

Weight - Buoyant force = ma

or 0.0245 - rho*V*g = ma

or 0.0245 - 1000*3.5*10^-7*9.81 = 2.5*10^-3*a

or a=8.4266 m/s^2

so let the time taken be T.so,

0.375 = uT + 0.5aT^2

or 0.375 = 2.08*T + 0.5*8.4266*T^2

or T=0.14 s

so total time =T+t

=0.14 + 0.239

=0.379 seconds