Rocket Acceleration in a Uniform Gravitational Field: Analyzing Momentum Change

• Gogsey
In summary, the conversation discusses the acceleration of a rocket in a uniform gravitational field, with notation for mass, initial mass, exhaust speed, and fuel consumption rate. By considering momentum changes, it is shown that mdv/dt = kv(ext) - mg. It is then stated that if k is constant, the speed v can be found as a function of remaining mass m. It is also mentioned that if k is very large, this result agrees with the one derived in class without gravity. The second part of the conversation discusses how to find v as a function of m, with a hint to assume that k is constant and v(ext) is constant as well.
Gogsey
A rocket is accelerating upwards from rest in a uniform gravitational field g. Notation: m(t) is the mass of the rocket plus remaining fuel, m0 is the initial total mass, vex is the exhaust speed (relative to the rocket), and k is the rate, in kg/s, at which fuel is consumed. By considering momentum changes in a short time dt, show that mdv/dt = kv(ext) - mg(upward direction is positive). Assuming k is constant, find the speed v as a function of the
remaining mass m. Show that, if k is very large, it agrees with the result derived in class (without gravity).

Ok, so i have already proved the first part.

P(t)=mv and P(t+dt) = (m+dm)(v+dv)-dM(v-v(ext))

=mv+vdm+dmv(ext)

The a couple of steps later we get: -mgdt=mdv+(ext)(dm/dt)

so we get -mg=mdv/dt -kv(ext), where -k=dm/dt

and therefore mdv/dt=kv(ext) - mg.

Ok, so I'm a little stuck on the second part. Not really sure where to start off. Any hints?

Assuming that k=dm/dt is constant isolate dv then integrate. You have to assume the exhaust velocity, v(ext) is constant.

1. How does acceleration affect a rocket's motion in a uniform gravitational field?

The acceleration of a rocket in a uniform gravitational field is dependent on the force of gravity acting on the rocket. As the rocket accelerates, the force of gravity remains constant, but the rocket's velocity and momentum increase.

2. What factors influence the acceleration of a rocket in a uniform gravitational field?

The acceleration of a rocket in a uniform gravitational field is influenced by the mass of the rocket, the force of gravity, and the direction and magnitude of the rocket's thrust.

3. How is momentum change calculated in the context of rocket acceleration in a uniform gravitational field?

Momentum change is calculated by multiplying the mass of the rocket by its change in velocity. This can be represented mathematically as Δp = mΔv, where Δp is the change in momentum, m is the mass of the rocket, and Δv is the change in velocity.

4. What is the significance of analyzing momentum change in rocket acceleration?

Analyzing momentum change in rocket acceleration allows us to understand the relationship between force, mass, acceleration, and velocity. It also helps us to predict the motion of a rocket and make calculations for successful launches.

5. How does the direction of a rocket's acceleration affect its momentum change?

The direction of a rocket's acceleration affects its momentum change by changing the direction of its velocity. If a rocket accelerates in the same direction as its velocity, its momentum will increase. However, if the acceleration is in the opposite direction, the rocket's momentum will decrease.

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