# The distance of closest approach of an alpha-particle fired towards a nucleus with momentum p is r. If the momentum of the alpha-particle is 2p, the corresponding distance of closest approach is (1) 4r (2) 2r (3) 2/r (4) r/4

All of the kinetic energy of incident particles is converted to the potential energy of the system at the distance of closest approach.

PE = KE

$$\begin{array}{l}\frac{p^{2}}{2m} = \frac{1}{4\pi \epsilon _{0}}\times \frac{q_{1}q_{2}}{r}\end{array}$$

Initially

$$\begin{array}{l}\frac{p^{2}}{2m}= \frac{2eq}{4\pi \epsilon _{0}r}\end{array}$$
………………………..(1)

Now

$$\begin{array}{l}\frac{2p^{2}}{2m}= \frac{2eq}{4\pi \epsilon _{0}r^{1}}\end{array}$$
………………….(2)

Equation (1) divided by (2)

1/4 = r1/r

r1 = r/4