The 2004 landings of the Mars rovers Spirit and Opportunity...

The 2004 landings of the Mars rovers Spirit and Opportunity involved many stages, resulting in each probe having zero vertical velocity about $12 \mathrm{~m}$ above the surface of Mars. Determine (a) the time required for the final free-fall descent of the probes and (b) the vertical velocity at impact. The mass of Mars is $6.419 \times 1023 \mathrm{~kg}, 6.419 \times 10^{23} \mathrm{~kg}$, and its radius is $3.397 \times 106 \mathrm{~m} 3.397 \times 10^{6} \mathrm{~m}$. Example $10-2$

Key Concepts

Energy Conservation in Free Fall

Energy conservation principles often provide an alternative method to analyze free fall. The loss in gravitational potential energy as an object falls is converted into kinetic energy, which can be used to calculate the impact velocity. This concept is essential for understanding how the potential energy difference over a short fall translates into the kinetic energy at impact.

Uniformly Accelerated Motion (Kinematics)

When an object is in free fall under a constant gravitational acceleration, its motion can be described using the kinematic equations. These equations relate the displacement, time, initial velocity, and final velocity of the object. In the context of the problem, they are used to determine the time taken for the descent and the impact velocity from a known height with initial velocity set to zero.

Newton's Law of Universal Gravitation

This concept describes how every mass exerts an attractive gravitational force on every other mass, proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It is crucial for calculating the gravitational force, and hence the acceleration due to gravity, on the Mars rovers as they approach the surface.

Gravitational Acceleration Calculation

Using Newton's law, the acceleration due to gravity at a planet’s surface is given by g = GM/r², where G is the gravitational constant, M is the mass of the planet, and r is its radius. This calculation provides the near-surface gravitational acceleration that can be assumed nearly constant over the small descent distance and is necessary for analyzing the free-fall motion.

Transcript

00:01 In this problem, we're looking at a space probe that's landing on the planet mars.

00:05 We're told that 12 meters above the surface that it has no vertical velocity, and we're asked how long does it take to travel that final 12 meters, and with what speed does it hit the martian surface width? and i've defined here all my variables.

00:24 We're looking for vb in the end, and ra is the radius of mars plus the 12 meters, but i'll call that delta y.

00:36 You'll see y do that in a little bit.

00:39 And our b is just our m, radius of the, it's on the surface.

00:44 Now, since we have delta y is 12 meters, the radius of mars is something times 10 to the 6th.

00:55 So i think that's kind of insignificant.

00:59 So since delta y is much less than the radius of mars, we can draw our picture like we've done in the past.

01:09 We have, here is our point a, ya is equal to yb plus delta y, va, va is equal to zero, and then here we have some ground level.

01:25 We're looking for vb, vb, is equal to the question mark, and this is some yb down here.

01:42 I'm not going to, i couldn't make this zero, and this 12, but there's a reason why i'm doing this.

01:48 And i'll make my plus y direction upward.

01:52 Now, with that said, you might say, well, okay, when i sell diagrams like this, i use kinematics.

01:58 Well, you're going to do the same thing here.

02:00 But to do kinematics, we need acceleration due to gravity.

02:04 So let's find that.

02:06 G, m prime, m over rm squared, is equal to m g.

02:15 And i'll put a little subscript m to remind us that it's for mars.

02:22 So this gives me that, and this defines, this relation defines the acceleration of gravity.

02:29 G .m is equal to 6 .67 times 10 to minus 11, newton, meter squared, kilogram squared, 6 .419 times 10 to 23rd, kilograms, that is the mass of mars and the radius 3 .397 times 10 to the 6 meters, and we must square that.

03:11 And this works out to be 3 .71 meters per second square.

03:21 So that is the acceleration into gravity.

03:24 Now we can just do a kinematics formula.

03:27 We're looking for the time first.

03:29 How long is it take to fall that 12 meters.

03:32 Y b, y, a, plus v -a -y -a -y, dot the t -a -b, minus one -half g -m, delta t -a -b squared.

03:52 V -a -y is zero, so this term is gone.

03:56 Y -b minus y -a -a is minus delta -y...