Billiard Ball A billiard ball moving at a speed of 2.2 m / s strikes an identical stationary ba

Billiard Ball A billiard ball moving at a speed of 2.2 m /...

Billiard Ball A billiard ball moving at a speed of $2.2 \mathrm{~m} / \mathrm{s}$ strikes an identical stationary ball a glancing blow. After the collision, one ball is found to be moving at a speed of $1.1 \mathrm{~m} / \mathrm{s}$ in a direction making a $60^{\circ}$ angle with the original line of motion. (a) Find the velocity of the other ball. (b) Can the collision be inelastic, given these data?

Key Concepts

Elastic vs Inelastic Collision

The distinction between elastic and inelastic collisions lies in the conservation of kinetic energy. In elastic collisions, both momentum and kinetic energy are conserved, whereas in inelastic collisions, only momentum is conserved while kinetic energy is partially lost. This concept is crucial when analyzing collision outcomes and determining the nature of the collision.

Conservation of Momentum

In collision problems, the total momentum of a system is conserved provided there are no external forces. This means that the sum of the momentum vectors before the collision is equal to the sum after. This principle is applied to each spatial component separately, which is particularly important in two-dimensional collisions.

Conservation of Kinetic Energy

This principle states that in an elastic collision, the total kinetic energy of the system remains the same before and after the collision. In scenarios where the collision is inelastic, some of the kinetic energy is transformed into other forms of energy, such as heat or sound, even though momentum is still conserved.

Vector Decomposition in Collisions

When dealing with collisions in two dimensions, it is essential to decompose velocity vectors into their components. This allows the application of conservation laws to each direction independently, facilitating the calculation of the resulting speed and direction of the objects involved.

Transcript

00:01 Hello.

00:02 Okay, to solve this, we need to use conservation of momentum, where we have momentum in the x direction and also the y direction.

00:11 Now the momentum in the y direction is going to give us, we're going to use that to find, well, we're actually going to use two equations and two unknowns.

00:21 So the momentum in the y direction has to equal zero, which means the momentum of our first ball, which is m, b1, sine of theta, which theta is 60 degrees, has to equal the momentum of the second ball, and we'll call it v2, sine of fee, where fee is whatever its angle is.

00:48 So these have to be equal to each other.

00:52 So we have two unknowns here.

00:55 We don't know this angle, and we don't know the speed.

00:57 We're going to start by figuring out the angle, because that's a little bit easier for me.

01:01 And so to do that we need to solve this in terms of v2 so that we can sub it into our second equation.

01:09 So let's just do that real quick.

01:12 The ms are going to cancel out because they are the same mass.

01:16 And the v2, sorry, v2 is going to be v1 sine of theta, which is 60 degrees over the sine of fee.

01:38 Now the x direction, we have mass of the cube all times its original speed, which i just called v.

01:47 And that's going to equal m v1 cosine of theta plus m v2 cosine of fee.

01:56 So we're going to substitute this in for v2...

Please give Ace some feedback