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?
(b) How long does this trip around Mars take?
The circular velocity required to orbit a planet derives from V = sqrt[µ/R] where µ = the gravitational constant of the planet and R = the radius(in feet) of the orbiting body around the planet.
For Mars,
µ = 1.512819x10^15 ft.^3/sec.^2
R = 2111 miles = 11,146,080 feet
Therefore, V = sqrt[1.512819x10^15/11,146,080] = 11,650 ft./sec. = 7,943mph.
The time, or period, to complete an orbit derives from
T = 2(Pi)sqrt[r^3/µ]
...= 2(Pi)sqrt[11,146,080^3/1.512819x10^15] = 6011sec. = 100.18 minutes.
To solve this problem, we can use the concepts of projectile motion and circular motion. Let's break down each part of the question:
(a) To find the muzzle speed of the package, we need to understand that the package needs to travel in a circular orbit around Mars. In order for this to happen, the centripetal force acting on the package must provide the necessary centripetal acceleration. Since the package is launched horizontally, the only force acting on it is gravity.
We are given that the free-fall acceleration on Mars is three-eighths that on Earth. On Earth, the acceleration due to gravity is approximately 9.8 m/s². Therefore, on Mars, the acceleration due to gravity is (3/8) * 9.8 m/s² = 3.675 m/s².
The centripetal force required for circular motion can be calculated using the formula: F = (m * v²) / r, where F is the centripetal force, m is the mass of the package, v is the speed of the package, and r is the radius of Mars.
Since the package returns to its original location, the radius of the circular orbit is equal to the circumference of Mars, which is approximately 2 * π * r, where r is the radius of Mars.
Setting the centripetal force equal to the force of gravity, we can write the equation:
(m * v²) / (2 * π * r) = m * g
Simplifying the equation, we get:
v = √ ((2 * π * r * g) / m)
Substituting the values, we can calculate the muzzle speed of the package.
(b) To find the time taken for the package to complete one orbit around Mars, we need to first calculate the period of the orbit. The period is the time taken for one complete revolution around the planet.
The formula to calculate the period of a circular orbit is: T = (2 * π * r) / v, where T is the period, r is the radius of the orbit, and v is the speed of the package.
Substituting the values, we can calculate the time taken for the trip around Mars.
Note: The radius of Mars may be required in both parts of the question, but it is not given in the problem statement. You would need to find the radius of Mars from a reliable source in order to obtain accurate results.