If "X" denotes final velocity,then I think that the answer will be in the ratio 1:4.Since acceleration is same&equal forces are applied,Only mass matters here and therefore the ratio of their masses is 1:4.Hence the answer is also in the ratio 1:4

Introduction:
When a force acts on an object, it can cause the object to accelerate and gain kinetic energy. The amount of kinetic energy gained by an object depends on its mass and the square of its velocity. In this scenario, we have two bodies of masses 4 kg and 16 kg initially at rest. We need to find the ratio of x for which the same force must act on both bodies to produce the same kinetic energy.

Calculating Kinetic Energy:
The kinetic energy of an object can be calculated using the formula:

K.E. = 0.5 * mass * velocity^2

Since both bodies are initially at rest, their initial velocities are zero.

For the first body with a mass of 4 kg, the kinetic energy is given by:

K.E.1 = 0.5 * 4 kg * (0 m/s)^2 = 0 J (Joules)

For the second body with a mass of 16 kg, the kinetic energy is given by:

K.E.2 = 0.5 * 16 kg * (0 m/s)^2 = 0 J (Joules)

As we can see, both bodies have zero kinetic energy initially.

Applying a Force:
Now, let's apply a force to both bodies to make them accelerate and gain kinetic energy. The force acting on each body will be the same.

The equation relating force, mass, and acceleration is:

Force = mass * acceleration

Since the force is the same for both bodies, the acceleration will be inversely proportional to their masses.

For the first body with a mass of 4 kg, the acceleration is given by:

acceleration1 = Force / mass1 = F / 4 kg

For the second body with a mass of 16 kg, the acceleration is given by:

acceleration2 = Force / mass2 = F / 16 kg

Equating Kinetic Energy:
To produce the same kinetic energy in both bodies, the final velocities of both bodies should be the same.

The final velocity can be calculated using the formula:

Final velocity = initial velocity + acceleration * time

Since the initial velocity is zero for both bodies, the final velocities will be equal to the product of acceleration and time.

For the first body, the final velocity is given by:

v1 = acceleration1 * x

For the second body, the final velocity is given by:

v2 = acceleration2 * (1 - x)

Here, x represents the ratio of the distance traveled by the first body to the total distance traveled by both bodies.

To equate the kinetic energies of both bodies, we can use the formula:

K.E.1 = K.E.2

0.5 * mass1 * v1^2 = 0.5 * mass2 * v2^2

Substituting the expressions for v1 and v2, we get:

0.5 * 4 kg * (acceleration1 * x)^2 = 0.5 * 16 kg * (acceleration2 * (1 - x))^2

Simplifying the equation, we have:

(acceleration1 * x)^2 = (acceleration2 * (1 - x))^2

Taking the square root of both sides, we get:

acceleration1 * x = acceleration2 * (1 - x)

Simplifying further