9781118230725.pdf

ation we would rewrite Eq. 9-77 for components along the xaxis as

m 1 v 1 im 1 v 1 fcosu 1 m 2 v 2 fcosu 2 , (9-79)

and along the yaxis as

(9-80)

We can also write Eq. 9-78 (for the special case of an elastic collision) in terms of
speeds:
(kinetic energy). (9-81)

Equations 9-79 to 9-81 contain seven variables: two masses,m 1 andm 2 ; three
speeds,v 1 i,v 1 f, and v 2 f; and two angles,u 1 andu 2. If we know any four of these
quantities, we can solve the three equations for the remaining three quantities.

1

2 m^1 v^1 i

(^2) ^1
2 m^1 v^1 f
(^2) ^1
2 m^2 v^2 f
2
0 m 1 v 1 fsinu 1 m 2 v 2 fsinu 2.
9-9 SYSTEMS WITH VARYING MASS: A ROCKET 241
Checkpoint 9
In Fig. 9-21, suppose that the projectile has an initial momentum of 6 kg m/s, a final
xcomponent of momentum of 4 kg m/s, and a final ycomponent of momentum of
3kg m/s. For the target, what then are (a) the final xcomponent of momentum
and (b) the final ycomponent of momentum?

9-9SYSTEMS WITH VARYING MASS: A ROCKET

After reading this module, you should be able to...

9.36Apply the first rocket equation to relate the rate at which
the rocket loses mass, the speed of the exhaust products rel-
ative to the rocket, the mass of the rocket, and the accelera-
tion of the rocket.

9.37Apply the second rocket equation to relate the change in

the rocket’s speed to the relative speed of the exhaust

products and the initial and final mass of the rocket.

9.38For a moving system undergoing a change in mass at a

given rate, relate that rate to the change in momentum.

●In the absence of external forces a rocket accelerates at an

instantaneous rate given by

RvrelMa (first rocket equation),

in which Mis the rocket’s instantaneous mass (including

unexpended fuel), Ris the fuel consumption rate, and vrelis

the fuel’s exhaust speed relative to the rocket. The term Rvrel

is the thrust of the rocket engine.

●For a rocket with constant Randvrel, whose speed

changes from vitovfwhen its mass changes from MitoMf,

vfvivrelln (second rocket equation).

Mi

Mf

Learning Objectives

Key Ideas

Systems with Varying Mass: A Rocket

So far, we have assumed that the total mass of the system remains constant.
Sometimes, as in a rocket, it does not. Most of the mass of a rocket on its launch-
ing pad is fuel, all of which will eventually be burned and ejected from the nozzle
of the rocket engine. We handle the variation of the mass of the rocket as the
rocket accelerates by applying Newton’s second law, not to the rocket alone but
to the rocket and its ejected combustion products taken together. The mass of this
system does notchange as the rocket accelerates.

Finding the Acceleration

Assume that we are at rest relative to an inertial reference frame, watching a
rocket accelerate through deep space with no gravitational or atmospheric drag