Introduction:

When a small body of mass m is located on a horizontal plane and acquires a horizontal velocity v, the motion of the body is opposed by the friction force between the body and the surface. This friction force does work on the body, and the rate at which this work is done is known as power.

Calculating the power:

The power developed by the friction force can be calculated using the formula:

P = F * v

where P is the power, F is the force of friction, and v is the velocity of the body.

Friction force:

The friction force between the body and the surface can be calculated using the formula:

F = μ * N

where F is the friction force, μ is the coefficient of friction, and N is the normal force.

Normal force:

The normal force is the force exerted by the surface on the body perpendicular to the surface. In this case, when the body is on a horizontal plane, the normal force is equal to the weight of the body, which can be calculated using the formula:

N = m * g

where N is the normal force, m is the mass of the body, and g is the acceleration due to gravity.

Substituting values:

Substituting the value of the normal force into the equation for friction force:

F = μ * m * g

Substituting the value of the friction force into the equation for power:

P = (μ * m * g) * v

Mean power:

The mean power developed by the friction force during the whole time of motion can be calculated by dividing the total work done by the total time taken. In this case, the work done is the product of the force of friction and the displacement of the body.

Since the body is moving with a constant horizontal velocity, the displacement can be calculated using the formula:

s = v * t

where s is the displacement, v is the velocity, and t is the time.

Substituting the value of displacement into the equation for work done:

W = F * s

W = (μ * m * g) * (v * t)

The mean power can then be calculated by dividing the work done by the total time taken:

P_mean = W / t

P_mean = (μ * m * g) * (v * t) / t

P_mean = μ * m * g * v

Conclusion:

The mean power developed by the friction force during the whole time of motion is given by the equation P_mean = μ * m * g * v, where μ is the coefficient of friction, m is the mass of the body, g is the acceleration due to gravity, and v is the velocity of the body.