# Of course...the fuel burns off

**Classical Mechanics**Level 4

The US space shuttle used booster rockets attached to the side to launch into space. Upon ignition, the booster rockets provided a thrust force of 30 million Newtons while the initial mass M of the space shuttle is approximately 2 million kg. At launch then, if we consider the shuttle as going straight up, the net force on the shuttle is the thrust force up minus the gravitational force down, i.e.

\(\vec{F}_{Net}=\vec{F}_{Thrust} - \vec{F}_g=3 \times 10^7- 2 \times 10^6~g \)

which yields an initial acceleration on the launch pad of \(\vec{a}=\vec{F}_{Net}/M= 5.2~m/s^2 \hat{y}\).

While thrust force remains approximately constant over the first few minutes of flight, the acceleration of the shuttle is not constant as the booster rockets lose mass as they burn fuel. The rockets lose approximately 10,000 kg/s of mass (from the initial mass of \(2 \times 10^6~kg\)) and therefore the acceleration actually is above the initial \(5.2~m/s^2\). Given this mass loss rate, what is the acceleration of the shuttle after 30 seconds of flight? You may treat the shuttle's motion as perfectly vertical and neglect air resistance (although both are important in real shuttle launches).

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