An α - particle and a proton are accelerated through same potential difference.Find the ratio (vα/ vp)of velocities acquired by two particles

Sure, let's solve this step by step.

Step 1: Understand the given information We know that an α - particle is a helium nucleus with a charge of +2e and a mass of 4u. A proton has a charge of +e and a mass of 1u. Both particles are accelerated through the same potential difference.

Step 2: Use the formula for kinetic energy The kinetic energy (K.E.) acquired by a charged particle when accelerated through a potential difference (V) is given by K.E. = qV, where q is the charge of the particle.

Step 3: Express kinetic energy in terms of mass and velocity We also know that K.E. = 1/2 mv^2, where m is the mass of the particle and v is its velocity.

Step 4: Equate the two expressions for kinetic energy Setting qV = 1/2 mv^2, we can solve for v to get v = sqrt(2qV/m).

Step 5: Find the ratio of velocities The ratio of the velocities of the α - particle (vα) and the proton (vp) is then vα/vp = sqrt[(2qαV/mα) / (2qpV/mp)].

Step 6: Substitute the known values Since the α - particle has twice the charge and four times the mass of the proton, and both particles are accelerated through the same potential difference, we can substitute qα = 2qp and mα = 4mp into the equation to get vα/vp = sqrt[(2*2qpV/4mp) / (2qpV/mp)] = sqrt[1/2].

So, the ratio of the velocities of the α - particle and the proton is vα/vp = sqrt[1/2] or vα/vp = 1/sqrt[2].