Question

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 (4He) from gold-197 nuclei (197Au). The energy of the incoming helium nucleus was 7.78 ✕ 10−13 J, and the masses of the helium and gold nuclei were 6.68 ✕ 10−27 kg and 3.29 ✕ 10−25 kg, respectively (note that their mass ratio is 4 to 197). (Assume that the helium nucleus travels in the +x direction before the collision.)

(a)

If a helium nucleus scatters to an angle of 130° during an elastic collision with a gold nucleus, calculate the helium nucleus' final speed (in m/s) and the final velocity (magnitude in m/s and direction counterclockwise from the +x-axis) of the gold nucleus.

⁴He speed _________ m/s

¹⁹⁷Au velocity_________ m/s

¹⁹⁷Au direction__________ ° counterclockwise from the +x-axis

(b)

What is the final kinetic energy (in J) of the helium nucleus?

_________ J

Answer 1

(a)

The expression to find the kinetic energy is,

Rearrange the expression for velocity.

Here, is kinetic energy, is mass and is initial velocity of Helium.

Substitute, for and for in the expression of velocity,

As collision is elastic; apply conservation of momentum along x-axis,

Here, are the masses of helium and gold respectively, is initial velocity of helium, are scattered angle for helium and gold respectively, is the final velocity of gold and is the final velocity of gold.

The expression for conservation of momentum along y-axis is:

Rearranging the above expression of conservation of momentum respectively,

Take square of both of them respectively and add the expressions,

Now, from the trigonometric relation:

Use trigonometric relation in the above expression; the new expression is,

Now, the conservation of internal kinetic energy gives:

Rearrange the expression for

Combine the expressions of and solve,

Substitute, for, 130° for, for and for in the above expression,

Solve the quadratic equation and find the velocity of helium nuclei,

Solve the above expression and consider positive value,

Substitute, for, for, for, and for in expression and find the velocity of gold nuclei,

Hence the velocity of gold nuclei is.

Substitute for, for, for, for and for in expression of momentum along y-axis and find the angle of final velocity of gold nucleus,

Further solve,

Hence the angle of the velocity of gold nuclei is clockwise that is down the horizontal.

B)The expression to find the final kinetic energy of helium is,

Substitute, for and for in the expression of final kinetic energy of helium,

Hence the final kinetic energy of helium nucleus is.