(a) Asteroids have average densities of about 2500 kg / m^3 and radii from 470 km down to less t

(a) Asteroids have average densities of about 2500 kg / m^3...

Key Concepts

Density and Mass-Volume Relationship

Density is a measure of how much mass is contained in a given volume and is defined by the equation ? = M/Volume. For spherical objects, the volume is given by (4/3)?R³. This relationship is critical in astrophysics for calculating the overall mass of planets, moons, or asteroids from their observable dimensions and for comparing different celestial objects. It also plays a role in deriving gravitational properties by linking the mass of the object to its size and density.

Gravitational Acceleration and Surface Gravity

Gravitational acceleration at the surface of a celestial body, often referred to as surface gravity, is determined by the mass of the body and its radius, and it is calculated using the formula g = GM/R². This relationship is a key concept in astrophysics because it links the observable gravitational effects to the underlying mass distribution. Understanding this concept allows for the estimation of a celestial body's bulk properties, such as its average density, from measurements of surface gravity.

Spherically Symmetric Mass Distribution

A spherically symmetric mass distribution assumes that the mass of an object, such as a planet or asteroid, is evenly distributed throughout a sphere. This assumption simplifies gravitational calculations by allowing one to treat the entire mass as if it were concentrated at a single point at the sphere’s center. Such symmetry is foundational in deriving formulas for gravitational fields and escape velocities, as well as for relating the gravitational acceleration at the surface to the overall mass and size of the body.

Escape Velocity

Escape velocity is the minimum speed required for an object to overcome the gravitational pull of a celestial body without further propulsion. It is derived from equating the kinetic energy of the object to the gravitational potential energy and is given by the formula v = ?(2GM/R), where G is the gravitational constant, M is the mass of the body, and R is its radius. This concept is crucial when determining if a person or object can 'jump' off from the surface of a smaller celestial body and escape its gravity.

Transcript

00:01 In this question, we are told that asteroids have average densities of about 2 ,500 kilograms per meter cubed and radii from 470 kilometers down to less than a kilometer.

00:15 And we're asked to estimate the radius of the largest asteroid from which you could escape simply by jumping off, assuming that the asteroid has a spherically symmetric mass distribution.

00:29 And then in part b, we're asked to calculate something a little bit different.

00:34 So let's go ahead and start off with part a.

00:38 And we were given a hint.

00:40 The hint is that we can estimate our jump speed by relating it to the maximum height that you can jump on earth.

00:47 So in order to estimate the largest asteroid that we could escape from, we need to know the biggest escape speed that we can generate with our legs, right? so this is what this hint is about, right? so we're looking at, you know, a person and they jump off the ground with some speed.

01:17 That speed is going to be the escape speed.

01:19 And on earth, that speed is directly related to, you know, how high you can jump, right? so maybe you jump with a speed v and you reach a height h.

01:31 And so if we can estimate a height that we can typically jump, then we can use conservation of energy to estimate what that speed was that generated that height.

01:45 So think about what a typical height would be that somebody could maybe jump.

01:51 I think, you know, if you investigate this on your own, you know, how height you could jump.

01:57 Most people are going to be, you know, a foot off the ground, maybe two feet.

02:02 So i'm going to estimate that as about 30 centimeters or 0 .3 meters.

02:10 And what speed do you need on earth to generate that height, right? how fast you need to be leaving the ground in order to get to that height? we can calculate that using conservation of energy, right? because all of the kinetic energy at the beginning of the motion should be transmitted into gravitational potential at the height of the motion.

02:37 So the kinetic energy is one -half mv squared and the gravitational potential is mgh so we can just cancel off our m's multiply by two and take the square root.

02:54 Okay so here we have our estimate for this speed, the maximum speed that someone can generate with their leg muscles.

03:04 So we're just going to plug everything in here.

03:08 And from that calculation, i get about 2 .43 meters per second.

03:14 So answers are going to vary here, depending on what you estimate for that height.

03:20 So now we're going to use this as the escape speed.

03:24 We're going to assume that this is the largest escape speed or the largest speed that someone can generate and see how that corresponds.

03:33 Responds to the mass of the asteroid that can be escaped from.

03:39 So let's look at the formula for escape speed.

03:45 So that's going to be equal to 2gm over r.

03:51 And you can take a look at example 12 .5 in the book if you want to know more about this formula.

03:58 The problem with this formula is that we were given an average density for a particular or sorry, we were given an average density for asteroids, but we weren't given a radius or a mass.

04:19 In fact, the mass is what we're trying to, or the radius, sorry, is what we're trying to find.

04:25 So what we need to do is rewrite this formula to get rid of mass because that's something that we're not calculating and is not given to us.

04:34 And what we can do is we can rewrite the mass in terms of the density, right? so we know that mass is equal to density times volume.

04:46 And we can rewrite volume as 4 pi over 3, r cubed.

04:56 So the mass becomes 4 pi over 3, row r cubed.

05:02 And so we can just substitute that into the formula to get a new formula for escape speed.

05:09 So the 4 pi becomes 8 pi when it gets multiplied, by 2.

05:14 We've still got our g there...

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