Question

A 70 kg astronaut floating in space in a 110 kg MMU (manned maneuvering unit) experiences an acceleration of 0.029 m/s² when he fires one of the MMU’s thrusters.

1. If the speed of the escaping N₂ gas relative to the astronaut is 490 m/s, how much gas is used by the thruster in 5.0 s?
2. What is the thrust of the thruster?

Short answer

1. By the thruster 0.053 kg of gas used.
2. The thrust of the thruster is 5.19 N.

Step-by-step solution

The given data

Given that a 70 kg astronaut floating in space in a 110 kg MMU (manned maneuvering unit) experiences an acceleration of 0.029 m/s² when he fires one of the MMU’s thrusters.

Mass of astronauts,$\mathrm{m}=70\mathrm{kg}$

Mass of unit, $\mathrm{M}=110\mathrm{kg}$

Acceleration, $\mathrm{a}=0.029\mathrm{m}/{\mathrm{s}}^2$

Concept used

In rocket propulsion, the mass of a rocket changes as the fuel is used up and ejected from the rocket. Analysis of the motion must contain the momentum carried away by the spent fuel as well as the momentum of the rocket itself.

(a) Find the gas used by the thruster.

Let $\frac{d\mathrm{m}}{d\mathrm{t}}$is mass lost as the rocket accelerates and v is the exhaust speed.

Hence, thrust can be found as,

$\mathrm{F}=-\mathrm{v}\frac{d\mathrm{m}}{d\mathrm{t}}$ ….. (1)

Calculate the total weight as below.

$\begin{array}{rcl}\mathrm{W}& =& 110+70\\ & =& 180\mathrm{kg}\end{array}$

The speed, $\mathrm{v}=490\mathrm{m}/\mathrm{s}$

Determine the lost mass as given below.

$\begin{array}{rcl}\frac{d\mathrm{m}}{d\mathrm{t}}& =& -\frac{\mathrm{m}}{\mathrm{v}}\mathrm{a}\\ & =& -\left(\frac{180}{490}\right)\left(0.029\right)\\ & =& -0.0106\mathrm{kg}/\mathrm{s}\end{array}$

Loss of mass in 5 seconds is,

$\begin{array}{rcl}\mathrm{m}& =& 0.0106\mathrm{kg}/\mathrm{s}\times 5\mathrm{s}\\ & =& 0.053\mathrm{kg}\end{array}$

(b) Find the thrust of the thruster

Thrust of the thruster is define by substituting known values into equation (1).

$\begin{array}{rcl}\mathrm{F}& =& -\mathrm{v}\frac{d\mathrm{m}}{d\mathrm{t}}\\ & =& -\left(490\right)\left(-0.0106\right)\\ & =& 5.19\mathrm{N}\end{array}$

Hence, the thrust of the thruster is $5.19\mathrm{N}$.