Week 6: Continuous Mass Transfer

A rocket sled can eject gas backwards or forwards at a speed \(\displaystyle u\) relative to the sled. The initial mass of the fuel in the sled is equal to dry mass of the sled, \(\displaystyle m_0\). At \(\displaystyle t=0\) the rocket sled has speed \(\displaystyle v_0\) and starts to eject fuel in the forward direction in order to slow down. You may ignore air resistance. You may treat \(\displaystyle u\) as a given constant in the following questions.

(Part a) Let \(\displaystyle v_ r(t)\) be the x-component of the rocket sled velocity, and \(\displaystyle m_ r(t)\) the mass of the rocket, dry mass and the fuel inside the rocket, at a given time \(\displaystyle t\). Derive differential equation for \(\displaystyle v_ r\) in terms of \(\displaystyle m_ r\).

(Part b) Integrate the equation you derived for \(\displaystyle v_ r\) to find the velocity of the rocket sled as a function of mass, \(\displaystyle v_ r(m_ r)\), as the rocket sled slows down. Write your answer using some or all of the following: \(\displaystyle u\), \(\displaystyle v_0\),\(\displaystyle m_0\), and \(\displaystyle m_ r\).

(Part c) What was the initial speed \(\displaystyle v_0\) of the rocket sled if the sled came to rest just as all the fuel was burned? Write your answer using some or all of the following: \(\displaystyle u\), \(\displaystyle m_0\), and \(\displaystyle m_ r\).

Rocket Sled - Differential Equation

View video page

Rocket Sled - Integrate the Rocket Equation

View video page

Rocket Sled - Solve for Initial Velocity

View video page

« Previous | Next »

Snow of density \(\displaystyle \rho\) covers a road to a uniform depth of \(\displaystyle D\) meters. A snowplowing truck of mass \(\displaystyle M\) starts clearing the road at \(\displaystyle t = 0\) at an initial velocity \(\displaystyle v_0\). The contact between the tires and the road applies a constant force \(\displaystyle F_0\) in the forward direction. The truck’s subsequent velocity depends on time as it clears a path of width \(\displaystyle W\) through the snow. The snow, after coming momentarily to rest relative to the truck, is ejected sideways, perpendicular to the truck.

(Part a) Find a differential equation relating the change in the velocity of the truck \(\displaystyle dv/dt\) to its velocity \(\displaystyle v(t)\). Express you answer in terms of some or all of the following: \(\displaystyle \rho\), \(\displaystyle D\), \(\displaystyle W\), \(\displaystyle M\), \(\displaystyle v\) and \(\displaystyle F_0\).

(Part b) Calculate \(\displaystyle v_{\text {term}}\), the terminal speed reached by the truck. Express you answer in terms of some or all of the following: \(\displaystyle \rho\), \(\displaystyle D\), \(\displaystyle W\), \(\displaystyle M\) and \(\displaystyle F_0\).

View video page