An alpha particle (a helium nucleus, containing 2 protons and 2 neutrons) starts out with kinetic energy of 10 MeV (10 ×
106 eV), and heads in the +x direction straight toward a gold nucleus (containing 79 protons and 118 neutrons). The particles
are initially far apart, and the gold nucleus is initially at rest. Answer the following questions about the collision. If you do
Problem 10.P.38, this problem will help you get started.
(a) What is the initial momentum of the alpha particle? (You may assume that its speed is small compared to the speed
of light).
(b) What is the final momentum of the alpha particle, long after it interacts with the gold nucleus?
(c) What is the final momentum of the gold nucleus, long after it interacts with the alpha particle?
(d) What is the final kinetic energy of the alpha particle?
(e) What is the final kinetic energy of the gold nucleus?
(f) Assuming that the movement of the gold nucleus is negligible, calculate how close the alpha particle will get to the
gold nucleus in this head-on collision.
(g) What principle did you use to calculate the distance of closest approach?
(h) The radius of the alpha particle is about 2 × 10−15 m, and the radius of the gold nucleus is about 8 × 10−15 m. At
the point of closest approach, do the two particles touch each other?
(i) A gold nucleus contains 197 nucleons packed tightly against each other. A single nucleon (proton or neutron) has a
radius of about 1.3 × 10−15 m. Calculate the approximate radius of the gold nucleus. (Hint: Remember that the
volume of a sphere is .)
(j) Also calculate the approximate radius of the alpha particle, which consists of 4 nucleons.
(k) To choose initial conditions for a program you may write to model collisions between an alpha particle and a gold
nucleus, you will need to choose an initial location, and Δt.
(l) Estimating Δt: What is the initial speed of the alpha particle?
(m) We suggest that in a computational model of this process that you start the alpha particle at a distance of 1 × 10−13
m from the gold nucleus, because when it is much farther away than this, its trajectory is nearly a straight line at
constant speed. If the alpha particle did not slow down, how long would it take to travel all the way to the location
of the nucleus (a distance of 1 × 10−13 m)?
For an accurate computer calculation you would need to take many steps during the time the alpha particle is near
the gold nucleus. Considering your estimate above, we suggest that you use Δt = 1 × 10−23 s in a computational
model.