Question

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7 A rocket of mass 10000 kg uses 5.0kg of fuel and oxygen
to produce exhaust gases ejected at 5000 m/s. Calculate the
increase in its velocity

Answer 1

Approximately
`2.5\; \rm m\cdot s^(-1)`, assuming that no external force (e.g., gravitational pull) was acting on this rocket.

Step-by-step explanation:

Assume that no external force is acting on this rocket. The system of the rocket and the fuel on the rocket would be isolated (an isolated system.) The momentum within this system would be conserved.

Let
`v_(0)\; \rm m\cdot s^(-1)` be the initial velocity of the rocket.

The velocity of the exhaust gas would be
`(v_(0) - 5000)\; \rm m\cdot s^(-1)` since the gas is ejected away from the rocket.

Let
`\Delta v\; \rm m\cdot s^(-1)` denote the increase in the velocity of the rocket. The velocity of the rocket after ejecting the gas would be
`(v + \Delta v)\; \rm m\cdot s^(-1)`.

The momentum
`p` of an object of velocity
`v` and mass
`m` is
`p = m \cdot v`.

The combined mass of the rocket and the fuel was
`10000\; \rm kg`. The initial momentum of this rocket-fuel system would be:

`\begin{aligned}p_(0) &= m \cdot v\\ &= 10000\; {\rm kg} * v_(0)\; {\rm m \cdot s^(-1)} \\ &= (10000\; v_(0))\; \rm {kg \cdot m\cdot s^(-1)}\end{aligned}`.

The momentum of the
`5.0\; \rm kg` of fuel ejected at
`(v_(0) - 5000)\; \rm m\cdot s^(-1)` would be:

`\begin{aligned} & 5.0 \; {\rm kg} * (v_(0) - 5000)\; {\rm m\cdot s^(-1)}\\ =\; & (5.0\, v_(0) - 25000)\; {\rm kg \cdot m \cdot s^(-1)}\end{aligned}`.

After ejecting the
`5.0\; \rm kg` of the fuel, the mass of the rocket would be
`10000\; \rm kg - 5.0\; \rm kg = 9995\; \rm kg`. At a velocity of
`(v + \Delta v)\; \rm m\cdot s^(-1)`, the momentum of the rocket would be:

`\begin{aligned} & 9995 \; {\rm kg} * (v_(0) + \Delta v)\; {\rm m\cdot s^(-1)}\\ =\; & (9995\, v_(0) + 9995\, \Delta v)\; {\rm kg \cdot m \cdot s^(-1)}\end{aligned}`.

Take the sum of these two quantities to find the momentum of the rocket-fuel system after the fuel was ejected:

`\begin{aligned}p_(1) &= (5.0\, v_(0) - 25000)\; {\rm kg \cdot m\cdot s^(-1)\\ &\quad\quad + (9995\, v_(0) + 9995\, \Delta v)\; {\rm kg \cdot m \cdot s^(-1)} \\ &= (10000\, v_(0) + 9995\, \Delta v - 25000)\; {\rm kg \cdot m \cdot s^(-1)}\end{aligned}`.

The momentum of the rocket-fuel system would be conserved. Thus
`p_(0) = p_(1)`.

`(10000\, v_(0))\; {\rm kg \cdot m\cdot s^(-1)} = (10000\, v_(0) + 9995\, \Delta v - 25000)\; {\rm kg \cdot m \cdot s^(-1)}`.

Solve this equation for
`\Delta v`, the increase in the velocity of the rocket.

`10000\, v_(0) = 10000\, v_(0) + 9995\, \Delta v - 25000`.

`9995\, \Delta v = 25000`.

`\begin{aligned}\Delta v &= (25000)/(9995) \approx 2.5\end{aligned}`.

Thus, the velocity of the rocket would increase by approximately
`2.5\; \rm m\cdot s^(-1)` after ejecting the
`5.0\; \rm kg` of fuel.