# 9.7 Rocket propulsion  (Page 4/8)

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## Rocket in a gravitational field

Let’s now analyze the velocity change of the rocket during the launch phase, from the surface of Earth. To keep the math manageable, we’ll restrict our attention to distances for which the acceleration caused by gravity can be treated as a constant g .

The analysis is similar, except that now there is an external force of $\stackrel{\to }{F}=\text{−}mg\stackrel{^}{j}$ acting on our system. This force applies an impulse $d\stackrel{\to }{J}=\stackrel{\to }{F}dt=\text{−}mgdt\stackrel{^}{j}$ , which is equal to the change of momentum. This gives us

$\begin{array}{ccc}\hfill d\stackrel{\to }{p}& =\hfill & d\stackrel{\to }{J}\hfill \\ \hfill {\stackrel{\to }{p}}_{\text{f}}-{\stackrel{\to }{p}}_{\text{i}}& =\hfill & \text{−}mgdt\stackrel{^}{j}\hfill \\ \hfill \left[\left(m-d{m}_{g}\right)\left(v+dv\right)+d{m}_{g}\left(v-u\right)-mv\right]\stackrel{^}{j}& =\hfill & \text{−}mgdt\stackrel{^}{j}\hfill \end{array}$

and so

$mdv-d{m}_{g}u=\text{−}mgdt$

where we have again neglected the term $d{m}_{g}dv$ and dropped the vector notation. Next we replace $d{m}_{g}$ with $\text{−}dm$ :

$\begin{array}{ccc}\hfill mdv+dmu& =\hfill & \text{−}mgdt\hfill \\ \hfill mdv& =\hfill & \text{−}dmu-mgdt.\hfill \end{array}$

Dividing through by m gives

$dv=\text{−}u\frac{dm}{m}-gdt$

and integrating, we have

$\text{Δ}v=u\phantom{\rule{0.2em}{0ex}}\text{ln}\left(\frac{{m}_{\text{i}}}{m}\right)-g\text{Δ}t.$

Unsurprisingly, the rocket’s velocity is affected by the (constant) acceleration of gravity.

Remember that $\text{Δ}t$ is the burn time of the fuel. Now, in the absence of gravity, [link] implies that it makes no difference how much time it takes to burn the entire mass of fuel; the change of velocity does not depend on $\text{Δ}t$ . However, in the presence of gravity, it matters a lot. The − g $\text{Δ}t$ term in [link] tells us that the longer the burn time is, the smaller the rocket’s change of velocity will be. This is the reason that the launch of a rocket is so spectacular at the first moment of liftoff: It’s essential to burn the fuel as quickly as possible, to get as large a $\text{Δ}v$ as possible.

## Summary

• A rocket is an example of conservation of momentum where the mass of the system is not constant, since the rocket ejects fuel to provide thrust.
• The rocket equation gives us the change of velocity that the rocket obtains from burning a mass of fuel that decreases the total rocket mass.

## Key equations

 Definition of momentum $\stackrel{\to }{p}=m\stackrel{\to }{v}$ Impulse $\stackrel{\to }{J}\equiv {\int }_{{t}_{\text{i}}}^{{t}_{\text{f}}}\stackrel{\to }{F}\left(t\right)dt\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{J}={\stackrel{\to }{F}}_{\text{ave}}\Delta t$ Impulse-momentum theorem $\stackrel{\to }{J}=\Delta \stackrel{\to }{p}$ Average force from momentum $\stackrel{\to }{F}=\frac{\Delta \stackrel{\to }{p}}{\Delta t}$ Instantaneous force from momentum (Newton’s second law) $\stackrel{\to }{F}\left(t\right)=\frac{d\stackrel{\to }{p}}{dt}$ Conservation of momentum $\frac{d{\stackrel{\to }{p}}_{1}}{dt}+\frac{d{\stackrel{\to }{p}}_{2}}{dt}=0\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}{\stackrel{\to }{p}}_{1}+{\stackrel{\to }{p}}_{2}=\text{constant}$ Generalized conservation of momentum $\sum _{j=1}^{N}{\stackrel{\to }{p}}_{j}=\text{constant}$ Conservation of momentum in two dimensions $\begin{array}{c}{p}_{\text{f},x}={p}_{\text{1,i},x}+{p}_{\text{2,i},x}\hfill \\ {p}_{\text{f},y}={p}_{\text{1,i},y}+{p}_{\text{2,i},y}\hfill \end{array}$ External forces ${\stackrel{\to }{F}}_{\text{ext}}=\sum _{j=1}^{N}\frac{d{\stackrel{\to }{p}}_{j}}{dt}$ Newton’s second law for an extended object $\stackrel{\to }{F}=\frac{d{\stackrel{\to }{p}}_{\text{CM}}}{dt}$ Acceleration of the center of mass ${\stackrel{\to }{a}}_{\text{CM}}=\frac{{d}^{2}}{d{t}^{2}}\left(\frac{1}{M}\sum _{j=1}^{N}{m}_{j}{\stackrel{\to }{r}}_{j}\right)=\frac{1}{M}\sum _{j=1}^{N}{m}_{j}{\stackrel{\to }{a}}_{j}$ Position of the center of mass for a system of particles ${\stackrel{\to }{r}}_{\text{CM}}\equiv \frac{1}{M}\sum _{j=1}^{N}{m}_{j}{\stackrel{\to }{r}}_{j}$ Velocity of the center of mass ${\stackrel{\to }{v}}_{\text{CM}}=\frac{d}{dt}\left(\frac{1}{M}\sum _{j=1}^{N}{m}_{j}{\stackrel{\to }{r}}_{j}\right)=\frac{1}{M}\sum _{j=1}^{N}{m}_{j}{\stackrel{\to }{v}}_{j}$ Position of the center of mass of a continuous object ${\stackrel{\to }{r}}_{\text{CM}}\equiv \frac{1}{M}\int \stackrel{\to }{r}\phantom{\rule{0.2em}{0ex}}dm$ Rocket equation $\Delta v=u\phantom{\rule{0.2em}{0ex}}\text{ln}\left(\frac{{m}_{\text{i}}}{m}\right)$

## Conceptual questions

It is possible for the velocity of a rocket to be greater than the exhaust velocity of the gases it ejects. When that is the case, the gas velocity and gas momentum are in the same direction as that of the rocket. How is the rocket still able to obtain thrust by ejecting the gases?

Yes, the rocket speed can exceed the exhaust speed of the gases it ejects. The thrust of the rocket does not depend on the relative speeds of the gases and rocket, it simply depends on conservation of momentum.

## Problems

(a) A 5.00-kg squid initially at rest ejects 0.250 kg of fluid with a velocity of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done in 0.100 s and there is a 5.00- N frictional force opposing the squid’s movement?

(b) How much energy is lost to work done against friction?

(a) 0.413 m/s, (b) about 0.2 J

two particles rotate in a rigid body then acceleration will be ?
same acceleration for all particles because all prticles will be moving with same angular velocity.so at any time interval u find same acceleration of all the prticles
Zaheer
what is electromagnetism
It is the study of the electromagnetic force, one of the four fundamental forces of nature. ... It includes the electric force, which pushes all charged particles, and the magnetic force, which only pushes moving charges.
Energy
what is units?
units as in how
praise
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F = m x a
Santos
State newton's second law of motion
can u tell me I cant remember
Indigo
force is equal to mass times acceleration
Santos
The acceleration of a system is directly proportional to the and in the same direction as the external force acting on the system and inversely proportional to its mass that is f=ma
David
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Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?
no, only the frequency and the material of the spring
Chun
Tech
beat line read important. line under line
Rahul
how can one calculate the value of a given quantity
means?
Manorama
To determine the exact value of a percent of a given quantity we need to express the given percent as fraction and multiply it by the given number.
AMIT
meaning
Winford
briefly discuss rocket in physics
ok let's discuss
Jay
What is physics
physics is the study of natural phenomena with concern with matter and energy and relationships between them
Ibrahim
a potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inductance of the inductor that should be connected to the capacitor for the circuit to oscillate at 1125Hza potential difference of 10.0v is connected across a 1.0AuF in an LC circuit. calculate the inducta
L= 0.002H
NNAEMEKA
how did you get it?
Favour
is the magnetic field of earth changing
what is thought to be the energy density of multiverse and is the space between universes really space
tibebeab
can you explain it
Guhan
Energy can not either created nor destroyed .therefore who created? and how did it come to existence?
this greatly depend on the kind of energy. for gravitational energy, it is result of the shattering effect violent collision of two black holes on the space-time which caused space time to be disturbed. this is according to recent study on gravitons and gravitational ripple. and many other studies
tibebeab
and not every thing have to pop into existence. and it could have always been there . and some scientists think that energy might have been the only entity in the euclidean(imaginary time T=it) which is time undergone wick rotation.
tibebeab
What is projectile?
An object that is launched from a device
Grant
2 dimensional motion under constant acceleration due to gravity
Awais
Not always 2D Awais
Grant
Awais
why not? a bullet is a projectile, so is a rock I throw
Grant
bullet travel in x and y comment same as rock which is 2 dimensional
Awais
components
Awais
no all pf you are wrong. projectile is any object propelled through space by excretion of a force which cease after launch
tibebeab
for awais, there is no such thing as constant acceleration due to gravity, because gravity change from place to place and from different height
tibebeab
it is the object not the motion or its components
tibebeab
where are body center of mass on present.
on the mid point
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is the magnetic field of the earth changing?
tibebeab
does shock waves come to effect when in earth's inner atmosphere or can it have an effect on the thermosphere or ionosphere?
tibebeab
and for the question from bal want do you mean human body or just any object in space
tibebeab