Ernest Rutherford (the first New Zealander to be awarded the...

Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei from gold-197 nuclei. The energy of the incoming helium nucleus was $8.00 \times 10^{-13} \mathrm{J},$ and the masses of the helium and gold nuclei were $6.68 \times 10^{-27} \mathrm{kg}$ and $3.29 \times 10^{-25} \mathrm{kg},$ respectively (note that their mass ratio is 4 to 197 ). a. If a helium nucleus scatters to an angle of $120^{\circ}$ during an elastic collision with a gold nucleus, calculate the helium nucleus's final speed and the final velocity (magnitude and direction) of the gold nucleus. b. What is the final kinetic energy of the helium nucleus?

Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei from gold-197 nuclei. The energy of the incoming helium nucleus was $8.00 \times 10^{-13} \mathrm{J},$ and the masses of the helium and gold nuclei were $6.68 \times 10^{-27} \mathrm{kg}$ and $3.29 \times 10^{-25} \mathrm{kg},$ respectively (note that their mass ratio is 4 to 197 ).
a. If a helium nucleus scatters to an angle of $120^{\circ}$ during an elastic collision with a gold nucleus, calculate the helium nucleus's final speed and the final velocity (magnitude and direction) of the gold nucleus.
b. What is the final kinetic energy of the helium nucleus?

Key Concepts

Elastic Collision

An elastic collision is a type of collision where both momentum and kinetic energy are conserved. This concept is central in physics when analyzing events where the interacting bodies rebound without deformation or loss of energy due to factors like friction or heat. In elastic collisions, the mathematical relationships derived from conservation laws allow us to predict final speeds and trajectories of the particles involved.

Conservation of Linear Momentum

The conservation of linear momentum principle states that the total momentum of a closed system remains constant if no external forces are acting on it. This concept is applied in analyzing collisions, and it involves setting up equations for each component direction (usually x and y in two-dimensional collisions) to determine how individual particle momenta contribute to the overall momentum before and after a collision.

Conservation of Kinetic Energy

In certain types of collisions, particularly elastic collisions, the total kinetic energy of the system is conserved. This means that the sum of the kinetic energies of all objects before the collision is equal to that after the collision. This principle is used alongside momentum conservation to derive the relationships between the incoming and outgoing speeds of the interacting particles.

Two-Dimensional Collision Analysis

Analyzing collisions in two dimensions requires breaking down velocities and momenta into their respective perpendicular components. This approach allows for the application of conservation laws separately in each direction, which is crucial when determining the outcomes, such as final speeds and directions, after an interaction. The use of trigonometric functions is often needed to relate the angles of deflection to these components.

Transcript

00:01 Okay, so we've got an alpha particle with energy of 8 times into the minus 13th joules.

00:06 That's its mass, and it strikes a gold nucleus, and that's the mass of the gold nucleus.

00:13 And the question is, what are their velocities afterwards? if, and then they tell us that this thing is scattered, and our angle here is 120 degrees, right? okay.

00:28 Now, if that's 120 degrees, then this here, if we try.

00:32 These components here, right? this guy is 30 degrees in here, right? okay.

00:42 And then right away, if we know that that's 30 degrees, some of the math we know, right, like the sign of 30 is one half, the cosine of 30 is square of three over two, right? so we know the relationship of that triangle right there.

00:57 Okay.

00:58 We also know that this is an elastic collision because it's basically it's the electrical repulsion of the electrons on the outside of this.

01:07 Actually, no, no, no.

01:08 It's the protons in this, actually repelling the protons on this, because this is a nuclear scattering.

01:16 So we know that the energy is going to be conserved.

01:21 So the first thing i want to do here actually is figure out what is this velocity.

01:24 We know this is the mass.

01:26 That's the energy, right? we know that kinetic energy is one -half mv squared, so therefore v is going to be the square root of twice the kinetic energy, so twice 8 .00 times 10 to the minus 13th, right, divided by the 6 .68 times 10 to the minus 27.

01:54 So let's figure that out.

01:55 We've got to figure out.

01:56 Okay.

01:57 I'll just call that guy v.

01:58 Okay, so i'm going to go square root of two times eight times 10 to the minus 13th divided by 6 .68 times 10 to the minus 27th.

02:14 And that is, wow, i wish this calculator gives me commas...

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