During a head-on collision in Rutherford's \alpha-particle scattering experiment, an \alph… — JAMB UTME

During a head-on collision in Rutherford's $\alpha$ -particle scattering experiment, an $\alpha$ -particle of initial speed $v$ approaches a stationary heavy nucleus. The distance of closest approach achieved by the $\alpha$ -particle is $r_0$ . If the initial speed of the $\alpha$ -particle is increased to $2v$ , what will be the new distance of closest approach in terms of $r_0$ ?

$\frac{r_0}{4}$ Answer
B
$\frac{r_0}{2}$
C
$2r_0$
D
$4r_0$

Answer

The new distance of closest approach will be

\frac{r_0}{4}

At the distance of closest approach, the entire initial kinetic energy of the

\alpha

-particle is converted into electric potential energy:

E_k = \frac{1}{2}mv^2 = \frac{k_e q_1 q_2}{r}

. Rearranging for

r

gives

r = \frac{2k_e q_1 q_2}{m v^2}

, showing that

r

is inversely proportional to

v^2

. When the initial speed is doubled to

2v

, the kinetic energy increases by a factor of

2^2 = 4

. Consequently, the distance of closest approach is reduced to one-fourth of its initial value,

\frac{r_0}{4}

Step-by-Step Solution

Apply conservation of mechanical energy at the distance of closest approach.

Initial kinetic energy equals electrostatic potential energy at distance

r_0

\frac{1}{2} m v^2 = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r_0}

At the distance of closest approach, the

\alpha

-particle momentarily stops, converting all kinetic energy into electric potential energy.

Express the distance of closest approach

r_0

in terms of initial speed

v

r_0 = \frac{2 q_1 q_2}{4\pi\varepsilon_0 m v^2} \propto \frac{1}{v^2}

Rearranging the energy conservation equation demonstrates that distance of closest approach is inversely proportional to

v^2

Substitute the new speed

v' = 2v

into the proportion.

r' \propto \frac{1}{(2v)^2} = \frac{1}{4v^2} = \frac{r_0}{4}

Doubling the speed quadruples the kinetic energy, reducing the distance required to bring the particle to rest by a factor of 4.

Key Concept

Distance of Closest Approach in Rutherford Scattering

Estimated Time:2m 0s

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