<< Chapter < Page Chapter >> Page >

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 F = m g j ^ acting on our system. This force applies an impulse d J = F d t = m g d t j ^ , which is equal to the change of momentum. This gives us

d p = d J p f p i = m g d t j ^ [ ( m d m g ) ( v + d v ) + d m g ( v u ) m v ] j ^ = m g d t j ^

and so

m d v d m g u = m g d t

where we have again neglected the term d m g d v and dropped the vector notation. Next we replace d m g with d m :

m d v + d m u = m g d t m d v = d m u m g d t .

Dividing through by m gives

d v = u d m m g d t

and integrating, we have

Δ v = u ln ( m i m ) g Δ t .

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

Remember that 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 Δ t . However, in the presence of gravity, it matters a lot. The − g 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 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 p = m v
Impulse J t i t f F ( t ) d t or J = F ave Δ t
Impulse-momentum theorem J = Δ p
Average force from momentum F = Δ p Δ t
Instantaneous force from momentum
(Newton’s second law)
F ( t ) = d p d t
Conservation of momentum d p 1 d t + d p 2 d t = 0 or p 1 + p 2 = constant
Generalized conservation of momentum j = 1 N p j = constant
Conservation of momentum in two dimensions p f , x = p 1,i , x + p 2,i , x p f , y = p 1,i , y + p 2,i , y
External forces F ext = j = 1 N d p j d t
Newton’s second law for an extended object F = d p CM d t
Acceleration of the center of mass a CM = d 2 d t 2 ( 1 M j = 1 N m j r j ) = 1 M j = 1 N m j a j
Position of the center of mass for a system
of particles
r CM 1 M j = 1 N m j r j
Velocity of the center of mass v CM = d d t ( 1 M j = 1 N m j r j ) = 1 M j = 1 N m j v j
Position of the center of mass of a
continuous object
r CM 1 M r d m
Rocket equation Δ v = u ln ( m i m )

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
what's microbiome?
john Reply
Microbiology is the scientific study of microorganisms
Ibra
the microorganisms in a particular environment (including the body or a part of the body).
Ibra
describe the bacterial cell
Akello
The biggest populations of microbes reside in the gut.other popular habitats include the skin genitals.The microbial cells and their genetic material ,the microbiome ,live with human from birth .
Zahreen
can agriculture be integrated into biology
David
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 1

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask