A helium atom (m=6.6465 ×10^-27 kg) collides elastically with an oxygen atom (m=2.6560 ×10^-26

A helium atom (m=6.6465 ×10^-27 kg) collides elastically...

A helium atom $\left(m=6.6465 \times 10^{-27} \mathrm{~kg}\right)$ collides elastically with an oxygen atom $\left(m=2.6560 \times 10^{-26} \mathrm{~kg}\right)$ at rest. After the collision, the helium atom is observed to be moving with a velocity of $6.636 \times 10^{6} \mathrm{~m} / \mathrm{s}$ in a direction at an angle of $84.7^{\circ}$ relative to its original direction. The oxygen atom is observed to move at an angle of $-40.4^{\circ} .(a)$ Find the speed of the oxygen atom. ( $b$ ) Find the speed of the helium atom before the collision.

Key Concepts

Vector Decomposition

In two-dimensional collision problems, velocity vectors are decomposed into their horizontal and vertical components. This process of vector resolution allows the application of conservation laws to each component independently, facilitating the establishment of equations that capture the dynamics of the system in multiple dimensions.

Conservation of Kinetic Energy

Conservation of kinetic energy is a key aspect of elastic collisions, ensuring that the total kinetic energy before the collision equals the total kinetic energy after the collision. This concept is used to set up an energy equation, which, together with momentum conservation equations, helps in solving for unknown velocities following the collision.

Conservation of Momentum

The law of conservation of momentum states that in the absence of external forces, the total momentum of a system remains constant throughout any interaction. In collisions, this principle is applied separately to each vector component (usually horizontal and vertical components in two-dimensional collisions), allowing us to set up equations that relate the initial and final momenta of the colliding objects.

Elastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved. This means that the total momentum vector of the colliding bodies before the collision is equal to the total momentum vector after the collision, and similarly, the sum of the kinetic energies of the bodies remains unchanged. This principle is fundamental in solving collision problems where the energy is not lost to deformation, sound, or heat.

Transcript

00:01 So in this example, they tell us that we have a helium atom and it collides with an oxygen atom at rest.

00:09 And we know that the mass of, or we know what the mass of the helium and the oxygen atom are.

00:15 They give us those.

00:17 We also know the final velocity of the helium atom.

00:21 And they tell us that the oxygen atoms at rest.

00:23 So we know that the initial velocity of the oxygen atom is going to be zero.

00:28 They also give us the deflection angles.

00:31 So we'll say that theta will represent the.

00:34 Angle at which the helium atom is deflected.

00:38 So they tell us that theta is equal to 84 .7 degrees.

00:41 And then phi will represent the angle the oxygen atom is deflected at, and that will equal negative 40 .4 degrees.

00:49 So to solve this problem, we're going to be using two equations.

00:55 It'll be the conservation momentum equation.

00:59 But because they're being deflected at different angles, we need one to represent the axis which will be equation one it will be their momentum along the x -axis and then we need another equation which will be equation two to represent their momentum along the y -axis because initially it is a horizontal collision we can see from equation two that their initial momentum along the y -axis is going to be zero so from there we will um start with solving part a of this problem which is where they ask what the final velocity of the oxygen atom is...

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