Two asteroids of equal mass in the asteroid belt between...

Name: Problem 31, Chapter 8: University Physics with Modern Physics
Uploaded: 2019-06-24T02:26:58-08:00
Duration: 8 min 1 s
Channel: Ben Nicholson

Two asteroids of equal mass in the asteroid belt between Mars and Jupiter collide with a glancing blow. Asteroid $A$, which was initially traveling at 40.0 m/s, is deflected 30.0$^\circ$ from its original direction, while asteroid $B$, which was initially at rest, travels at 45.0$^\circ$ to the original direction of $A$ ($\textbf{Fig. E8.31}$). (a) Find the speed of each asteroid after the collision. (b) What fraction of the original kinetic energy of asteroid $A$ dissipates during this collision? Figure 58.31 (CANT COPY FIGURE)

Key Concepts

Elastic and Inelastic Collision Dynamics

Collisions can be classified as elastic or inelastic. In an elastic collision, both momentum and kinetic energy are conserved, whereas in an inelastic collision, only momentum is conserved and some kinetic energy is lost. Analyzing whether a collision is elastic or inelastic is crucial for correctly applying the conservation laws and understanding the energy transformation that occurs during the collision.

Kinetic Energy and Its Dissipation

In collision problems, kinetic energy is sometimes not conserved, particularly in inelastic collisions. Comparing the total kinetic energy before and after the collision allows one to determine how much energy was converted to other forms, such as heat or deformation. This concept involves calculating the kinetic energies and then determining the fraction or percentage of energy that has been dissipated during the collision.

Conservation of Momentum

This principle states that in a closed system, the total momentum remains constant before and after a collision. In problems involving collisions, momentum must be conserved in all directions, which means that the momentum vector components (typically x and y components) are conserved separately. This allows us to set up equations based on the conservation of momentum along each axis to solve for unknown quantities after the collision.

Vector Decomposition in Two Dimensions

When dealing with collisions that occur at angles, it is necessary to break down the momentum vector into its perpendicular components using trigonometric functions such as sine and cosine. This decomposition simplifies the problem by allowing the analysis of the collision along independent axes, making it easier to apply conservation laws.

Transcript

00:01 Problem 8 .31, we have two asteroids colliding.

00:06 They have the same mass, which is very convenient.

00:11 The initial speed of this is 40 meters per second, and then they go off in two directions whose angles from the initial direction of motion are given.

00:23 And let's call these angles theta a and theta b.

00:31 And while we're at it, call this the positive x direction, and this the positive y direction.

00:47 So because we're not told of any forces acting external to the asteroids or any forces that there are are negligible because this is in the asteroid belt where everything is very far apart for the most part.

01:07 The momentum in both of these directions is going to be conserved.

01:11 So let's write down what we have for the x direction first.

01:18 So m, which is.

01:19 And has no subscript because they both have the same mass times va initial equals.

01:33 And now we're doing the x component, so we're going to want the cosines of these angles.

01:43 Cosine of theta a plus m cosine of theta b.

01:56 We'll note that all the m's cancel out here.

01:59 Now in the y direction there's initially zero momentum because this asteroid is only moving in what we're calling the x direction.

02:10 So then zero is going to be equal to m sine theta a.

02:20 Oh my apologies.

02:31 We do want the speeds easier.

02:39 Very important not to forget that.

02:46 Oh, it's even more important if you do forget that to then catch that before it's too late...

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