Explanation:

When a constant force acts on a mass m which is initially at rest, the velocity acquired in a given displacement is proportional to the square root of the mass. Mathematically, it can be represented as:

v ∝ √m

This relationship can be derived using Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force applied and inversely proportional to its mass. The equation for Newton's second law is:

F = ma

Where F is the net force, m is the mass, and a is the acceleration. In this case, we have a constant force acting on the mass, so we can rewrite the equation as:

F = constant × a

Since the mass is constant, the acceleration is also constant. Let's assume the acceleration is given by the symbol 'k'. Therefore, we can rewrite the equation as:

F = k × a

Now, we know that the acceleration is the rate of change of velocity with respect to time. If we integrate the acceleration with respect to time, we get the velocity. So, we can rewrite the equation as:

F = k × dv/dt

Where dv is the change in velocity and dt is the change in time. Rearranging the equation, we have:

dv = (F/k) × dt

Integrating both sides of the equation, we get:

∫dv = ∫(F/k) × dt

This simplifies to:

v = (F/k) × t + C

Where C is the constant of integration. Since we are considering the case where the mass is initially at rest, the initial velocity is zero. Therefore, C = 0. Hence, the equation becomes:

v = (F/k) × t

But we know that t = d/v, where d is the displacement. Substituting this in the equation, we get:

v = (F/k) × (d/v)

Simplifying, we have:

v^2 = (F/k) × d

Since the force F and the displacement d are constants, we can replace them with a constant 'C'. Therefore, the equation becomes:

v^2 = C

Taking the square root of both sides, we get:

v = √C

Hence, the velocity acquired in a given displacement is proportional to the square root of the mass (v ∝ √m). Therefore, the correct answer is option a) √m.

Option (a)