# Back-of-the-Envelope Calculations: Spacecraft Acceleration

#### Summary

*Question*

Suppose someone offered you a ride to the nearest star in a new spacecraft that could travel at half the speed of light, or about 150,000 km/second. In order to reach such a cruising speed, you and the spacecraft must accelerate from a standstill to half the speed of light. Acceleration means uncomfortable (and maybe even fatal!) "g" forces, that pressed-into-the-seat feeling you get when a car or airplane accelerates. More than 3 g's of acceleration are tough to take for very long, so your spacecraft's engines are designed to accelerate you at not more than 29 meters/second/second (3 times the acceleration due to gravity at the Earth's surface). How long will it take you and your spacecraft to accelerate to half light speed?

## Assessment

*Answer*

About 2 months. An acceleration of 29 m/sec/sec means that your speed will increase by 29 m/sec every second. After one second, your speed will be 29 m/sec; after two seconds, 58 m/sec; after three seconds, 87 m/sec. To complete the calculation, convert 150,000 km/sec to 150,000,000 meters per second! At an acceleration of 29 m/second every second, then, it takes 5,172,414 seconds to accelerate to half light speed. Divide that by the number of seconds in a day (86,400), and that gives you about 60 days. To accelerate to just half the speed of light without squashing everyone into a thin smear would take almost (yawn) two months! This puts into perspective all that rushing around from star system to star system in science fiction movies! Star Trek writers had to invent "inertial dampers" so that the crew wouldn't be turned into paté

## References and Resources

This SERC page describes the use of Back of the Envelope Calculations

A View from the Back of the Envelope (more info) : This site has a good number of easy simulations and visualizations of back of the envelope calculations.

The Back of the Envelope : This page outlines one of the essays in the book "Programming Pearls" (ISBN 0-201-65788-0). The book is written for computer science faculty and students, but this portion speaks very well to back of the envelope calculations in general.

## Controlled Vocabulary Terms

**Subject**: Mathematics, Geoscience:Lunar and Planetary Science

**Resource Type**: Activities:Classroom Activity:Short Activity

**Special Interest**: Quantitative

**Quantitative Skills**: Estimation

**Ready for Use**: Ready to Use

**Topics**: Solar system, Chemistry/Physics/Mathematics