An astronaut on the surface of Mars fires a cannon to launch an experiment package, which leaves...

Question:

An astronaut on the surface of Mars fires a cannon to launch an experiment package, which leaves the barrel moving horizontally. Assume that the free-fall acceleration on Mars is three eighths that on the Earth.(a) What must be the muzzle speed of the package so that it travels completely around Mars and returns to its original location?

Orbital Speed

An object of mass m can move around a planet of mass M in a very near orbit to the planet if the gravitational force between the planet and object is sufficient enough to provide centripetal force to the object. In that case the orbital speed of the object {eq}v = \sqrt { \dfrac { G M } { R } } = \sqrt { g R } {/eq}. In this equation G, g and R are the gravitational constant, acceleration due to gravity and radius of the planet respectively. So one can send an object around the planet by giving sufficient kinetic energy to the object.

Given data

• An experiment package is fired horizontally on the surface of the planet Mars
• Free fall acceleration on the surface of Mars {eq}g_m = \dfrac { 3 g } { 8 } {/eq}
• Acceleration due to gravity on the surface of earth {eq}g = 9.80 \ m/s^2 {/eq}
• Radius of the planet mars {eq}R = 3.390 \times 10^6 \ m {/eq}

The horizontally fired experimental package will move around the planet if the launch speed is equal to the orbital speed in the very close orbit almost equal to the radius of the planet.

Then in that case the required launch speed from the cannon firing or muzzle speed {eq}v = \sqrt { g_m R } \\ v = \sqrt { \dfrac { 3 g } { 8 } \times R } \\ v = \sqrt { \dfrac { 3 \times 9.80 \times 3.390 \times 10^6 } { 8 } } \\ v = 3529.6 \ m/s {/eq}

How Orbits Are Influenced by Gravity & Energy

from Basics of Astronomy

Chapter 25 / Lesson 6
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