How Does Rocket Propulsion Affect Velocity and Position in Physics Problems?

[winlinux]

6. A rocket is fired vertically upwards from rest and burns fuel at a constant rate k, and the exhaust gases are ejected vertically downwards with constant speed u relative to the rocket. The initial mass of the rocket is M', half of which is fuel. During the time that the fuel is being burned, air resistance may be neglected and the Earth’s gravitational field may be assumed to give rise to an acceleration of constant magnitude g, where ku >M'g. Show that the speed of the rocket when the fuel runs out is given by
uln2-gM'/2k, and find the distance traveled at this time.


2.According to Einstein’s theory of special relativity, the kinetic energy K of a mass m moving at velocity v is given by

K=[(mc^2)/(1-(v/c)^2)^0.5]-mc^2

Show that at v<<c, the kinetic energy of the mass reduced to the Newtonian expression.


5. (10 marks)
Two identical particles of mass m attract each other with a force that obeys Newton’s third law. They are initially at rest on a smooth inclined plane which has an angle of inclination of 45 degrees w.r.t. the horizontal. Find the acceleration of the center of mass of the system.

how to solve them?
thanks!

 

[TenaliRaman]

for 6,
Note that F = dp/dt where p is momentum

for 2,
do u know binomial expansion?
(note : v<<c implies v/c << 1)

for 5,
to me this seems equivalent to two balls held by a rod...
(i may be wrong ... its night time and sometimes my brain doesn't really work that well)

-- AI

 

[winlinux]

can anyone give me more hints?
thanks!

 

[arildno]

Can't you take one exercise at a time and write in detail your thoughts about it, and what you are uncertain about?
(This isn't just because we're lazy, but it will make you clarify for yourself as well your level of knowledge, and hence, what you need to become better at)

 

[winlinux]

in no.6
now i know 'u ln2' is come from v=uln(M'/0.5M')
but "-gM'/2k",how come can anyone explain to me?