Explanation:

When a particle moves on rough horizontal ground, friction comes into play. Friction is a force that opposes the motion of the particle, and it causes the particle to lose kinetic energy. In this scenario, let's assume the initial velocity of the particle is v.

Kinetic Energy:

The kinetic energy of a particle is given by the equation:

K.E. = 0.5 * m * v^2

where m is the mass of the particle and v is the velocity of the particle.

Loss of Kinetic Energy:

According to the problem statement, 3/4 of the kinetic energy is lost due to friction in time t. This means that the final kinetic energy of the particle is only 1/4 of the initial kinetic energy.

Let K.E.i be the initial kinetic energy and K.E.f be the final kinetic energy. Therefore, we have:

K.E.f = 1/4 * K.E.i

Substituting the formula for kinetic energy, we get:

0.5 * m * v_f^2 = 1/4 * (0.5 * m * v_i^2)

where v_f is the final velocity of the particle and v_i is the initial velocity of the particle.

Coefficient of Friction:

The coefficient of friction (μ) is a measure of the frictional force between two surfaces. It is defined as the ratio of the frictional force (F) to the normal force (N) between the surfaces.

In this case, the frictional force is responsible for the loss of kinetic energy. Therefore, we can express the frictional force as:

F = ΔK.E.

where ΔK.E. is the change in kinetic energy.

The normal force N is equal to the weight of the particle, which is given by:

N = m * g

where g is the acceleration due to gravity.

The frictional force can be written as:

F = μ * N

Substituting the expressions for N and F, we have:

μ * m * g = ΔK.E.

Solving for the Coefficient of Friction:

Now, we can substitute the expression for ΔK.E. obtained earlier:

μ * m * g = 0.5 * m * v_i^2 - 0.5 * m * v_f^2

Since we know that the final kinetic energy is 1/4 of the initial kinetic energy, we can substitute this relationship:

μ * m * g = 0.5 * m * v_i^2 - 0.5 * m * (1/4 * v_i^2)

Simplifying the equation, we get:

μ * g = 0.5 * v_i^2 - 0.5 * (1/4 * v_i^2)

μ * g = 0.5 * v_i^2 - 0.125 * v_i^2

μ * g = 0.375 * v_i^2

Finally, we can solve for the coefficient of friction μ:

μ = (0.375 * v_i^2) / g

The coefficient of friction between the particle and the ground is given by the expression (0.375 * v_i^2) / g.