Rocket/thrust problems (mass decreasing)

1 A rocket is launched vertically upwards from rest. The initial mass of the rocket, including fuel and payload, is \(m _ { 0 }\) and the propulsion system ejects mass at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. The only other force acting on the rocket is its weight. The acceleration due to gravity is constant throughout the motion. At time \(t\) after launch the speed of the rocket is \(v\).

1. Show that while mass is being ejected from the rocket \(v = u \ln \left( \frac { m _ { 0 } } { m _ { 0 } - k t } \right) - g t\). The rocket initially has 2400 kg of fuel which is ejected at a constant rate of \(100 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\) with constant speed \(3000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket.
2. Given that the rocket must reach a speed of \(7910 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before releasing its payload, find the maximum possible value of \(m _ { 0 }\).

4. A rocket-driven car propels itself forwards in a straight line on a horizontal track by ejecting burnt fuel backwards at a constant rate \(\lambda \mathrm { kg } \mathrm { s } ^ { - 1 }\) and at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the car. At time \(t\) seconds, the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the total resistance to the motion of the car has magnitude \(k v \mathrm {~N}\), where \(k\) is a positive constant. When \(t = 0\) the total mass of the car, including fuel, is \(M \mathrm {~kg}\). Assuming that at time \(t\) seconds some fuel remains in the car,

1. show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }$$
2. find the speed of the car at time \(t\) seconds, given that it starts from rest when \(t = 0\) and that \(\lambda = k = 10\).

5. A rocket is launched vertically upwards under gravity from rest at time \(t = 0\). The rocket propels itself upward by ejecting burnt fuel vertically downwards at a constant speed \(u\) relative to the rocket. The initial mass of the rocket, including fuel, is \(M\). At time \(t\), before all the fuel has been used up, the mass of the rocket, including fuel, is \(M ( 1 - k t )\) and the speed of the rocket is \(v\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { k u } { 1 - k t } - g\).
2. Hence find the speed of the rocket when \(t = \frac { 1 } { 3 k }\).

6. A rocket-driven car moves along a straight horizontal road. The car has total initial mass \(M\). It propels itself forwards by ejecting mass backwards at a constant rate \(\lambda\) per unit time at a constant speed \(U\) relative to the car. The car starts from rest at time \(t = 0\). At time \(t\) the speed of the car is \(v\). The total resistance to motion is modelled as having magnitude \(k v\), where \(k\) is a constant. Given that \(t < \frac { M } { \lambda }\), show that

1. \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }\),
2. \(v = \frac { \lambda U } { k } \left\{ 1 - \left( 1 - \frac { \lambda t } { M } \right) ^ { \frac { k } { \lambda } } \right\}\).
   (6)
   (Total 13 marks)

4. At time \(t = 0\) a rocket is launched from rest vertically upwards. The rocket propels itself upwards by expelling burnt fuel vertically downwards with constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket. The initial mass of the rocket is \(M _ { 0 } \mathrm {~kg}\). At time \(t\) seconds, where \(t < 2\), its mass is \(M _ { 0 } \left( 1 - \frac { 1 } { 2 } t \right) \mathrm { kg }\), and it is moving upwards with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { U } { ( 2 - t ) } - 9.8 .$$
2. Hence show that \(U > 19.6\).
3. Find, in terms of \(U\), the speed of the rocket one second after its launch.

5. At time \(t = 0\) a rocket is launched. The rocket has initial mass \(M\), of which mass \(\lambda M\), \(0 < \lambda < 1\), is fuel. The rocket is launched vertically upwards, from rest, from the surface of the Earth. The rocket burns fuel and the burnt fuel is ejected vertically downwards with constant speed \(U\) relative to the rocket. At time \(t\), the rocket has mass \(m\) and velocity \(v\). Ignoring air resistance and any variation in \(g\),

1. show, from first principles, that until all the fuel is used, $$m \frac { \mathrm {~d} v } { \mathrm {~d} t } + U \frac { \mathrm {~d} m } { \mathrm {~d} t } = - m g$$ The rocket accelerates vertically upwards with constant acceleration \(g\).
2. Show that \(m = M \mathrm { e } ^ { \frac { - 2 g t } { U } }\)
3. Find, in terms of \(M , U\) and \(\lambda\), an expression for the kinetic energy of the rocket at the instant when all of the fuel has been used.