Ch 31: Electromagnetic Fields and Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 31: Electromagnetic Fields and Waves

Problem 59a

Chapter 31, Problem 59a

It has been proposed that small spacecraft could reach other planets in a fairly short time—days instead of many months—if they unfurl a reflective sail and are accelerated by a powerful laser beam generated by an earth-orbiting laser. What speed would a spacecraft need to reach Mars in 8.0 days when Mars is closest to earth?

Verified step by step guidance

Determine the distance between Earth and Mars when Mars is at its closest approach to Earth. This distance is approximately 5.6 × 10^7 km (or 5.6 × 10^10 m).

Convert the given time of 8.0 days into seconds. Use the conversion factor: 1 day = 24 hours, 1 hour = 3600 seconds. Thus, time in seconds = 8.0 × 24 × 3600.

Use the formula for constant speed:

v = \frac{d}{t}

, where

v

is the speed,

d

is the distance, and

t

is the time. Substitute the values for

d

(5.6 × 10^10 m) and

t

(time in seconds from step 2).

Simplify the expression to calculate the required speed

v

. Ensure the units are consistent (meters per second).

Interpret the result: The calculated speed represents the constant velocity the spacecraft must maintain to reach Mars in 8.0 days when Mars is at its closest approach to Earth.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Acceleration and Force

Acceleration is the rate of change of velocity of an object, which is influenced by the net force acting on it according to Newton's second law (F=ma). In the context of the spacecraft, a powerful laser beam provides the necessary force to accelerate the spacecraft, allowing it to reach high speeds over a short period.

Distance and Speed

Speed is defined as the distance traveled per unit of time. To determine the speed required for the spacecraft to reach Mars in 8.0 days, one must calculate the distance to Mars when it is closest to Earth (approximately 54.6 million kilometers) and divide this distance by the time in hours (192 hours) to find the necessary speed in kilometers per hour.

Laser Propulsion

Laser propulsion involves using focused laser beams to impart momentum to a spacecraft, typically through a reflective sail. This method allows for continuous acceleration without the need for onboard fuel, making it a promising technology for rapid interplanetary travel, as it can achieve high speeds over time with minimal mass.

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