(50 points) A rocket of mass m is launched straight up with thrust Frust a) (3 points) Model: How will you model this problem? b) (6 points) Visualize: Draw a free body diagram of the forces acting on the rocket. (6 points) Visualize: Draw a pictorial representation of the rocket and unknown kinematic quantities. d) (15 points) Solve: Find the acceleration of the rocket in terms of known quantities m, thrust, and q e) (15 points) Solve: Find an expression for the rockets speed at height h if air resistance is neglected. (5 points)Assess: Verify that your expressions for parts d) and e) give you the correct dimensions of L/T2 and L/T, respectively. f)

a) To model this problem we can use the law of conservation of the moment where the system is formed by the rocket plus the ejected fuel that creates the propulsion

b) in the attached we can see a diagram of the free body the cohetees the aja grade and the fuel the small box

The ejected mass is m ’= Δm

c) we write the initial moment

p₀ = (m + Δm) v

Final moment

pf = m (v + Δv) + Δm (v- v ’)

Where v ’is the gas velocity relative to the rocket

p₀ = pf

(m + Δm) v = m (v + Δv) + Δm (v- v ’)

Δm v ’= m Δv

The amount Δm is negative because it is a decrease in rocket masses

Δv = -v’ Δm / m

dv = -v' dm/m

We integrate

v = v’ ln m

We evaluate between the lower limit v = 0 m = m0 and the upper limit v = v for m = m

v = v’ ln (m₀ / m)

d) acceleration is defined

a = dv / dt

a = dv / dm dm / dt

a = - v’/m dm/dt

The amount dm/dt in the amount of fuel burned per unit of time, in general is given for a specific engine

e) the rocket speed is found in part c

The position, we can find it from the define of the speed

v = dy / dt = dy /dm dm/dt

Dm / dt = C

dy = v dm / C

dy = v’/C ln (mo / m) dm

We integrate

y = v’/C [(m- m ln ( m)]

f) dimensional analysis

a = - v’/m dm/dt

[ m/s²] =[m/s]/[km] [km/s] = [ m/s²]

y = v’/C [(m- m ln ( m)]

[m] = [m/s]/[kg/s] [kg] = [m]