A rocket ascending vertically has variable mass and velocity for seconds

PHYSICS · Multi-part · A Level · Physics · Mechanics and materials · Mechanics and materials

Problem

A rocket ascending vertically has variable mass

m(t)=2000e^{-0.05t}\,\text{kg}

and velocity

v(t)=150\bigl(1-e^{-0.1t}\bigr)\,\text{m\!/s}

for

t\ge 0

seconds. The momentum of the rocket is

P(t)=m(t)\,v(t)

. Answer parts (a)–(e) with appropriate units. (a) Compute the exact momentum

P(10)

. Report both the exact form and a three–decimal–place approximation. (b) Obtain a simplified analytic expression for

P'(t)

. Show every differentiation step. (c) Evaluate

P'(10)

, include correct units, and interpret its physical meaning in the context of the rocket’s motion. (d) Using

P(10)

and

P'(10)

, write the equation of the tangent line to the graph of

P

t=10\,\text{s}

. (e) Use the tangent line from part (d) to estimate the rocket’s momentum at

t=11\,\text{s}

. Justify whether this estimate is likely an overestimate or an underestimate by discussing the concavity of

P

near

t=10

. Support your claim analytically.