Inside a rocket that is launching vertically upward

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Fluids · Fluids

Problem

Inside a rocket that is launching vertically upward, a small uniform sphere of density

\rho_s

floats in a tank of fuel (treated as an incompressible fluid of density

\rho_f>\rho_s

). During one phase of flight the rocket’s upward acceleration relative to an inertial frame is a constant

a_r

, with

a_r>0

but not necessarily equal to

g

. (a) Derive an expression for the fraction

f

of the sphere’s volume that is submerged while floating at rest relative to the accelerating rocket. Express

f

in terms of

\rho_s

\rho_f

g

, and

a_r

. (b) Evaluate

f

for a sphere of density

700\,\text{kg}\,\text{m}^{-3}

floating in a fluid of density

900\,\text{kg}\,\text{m}^{-3}

when

a_r=2.0\,\text{m}\,\text{s}^{-2}

. Show numerical steps. (c) Discuss how the submerged fraction changes as

a_r

varies from

-g

(rocket in free fall) to

+g

(rocket stationary on the launchpad). Identify any limiting values of

f

at these extremes and provide physical explanations. (d) Suppose the rocket’s engines fail and it enters free fall

\bigl(a_r=-g\bigr)

. Predict the subsequent motion, if any, of the sphere relative to the fluid, and justify your answer using your expression for

f