Given:
Mass of the body, m = 10 kg
Acceleration with a horizontal force of F newtons, a1 = 5 m/s^2
Acceleration with a horizontal force of 2F newtons, a2 = 18 m/s^2

Calculating the Coefficient of Friction:
To find the coefficient of friction between the body and the horizontal surface, we need to analyze the forces acting on the body.

1. Force due to Applied Force:
The force applied on the body is responsible for its acceleration. According to Newton's second law of motion, the force can be calculated using the formula:
F = ma

2. Force due to Friction:
The frictional force acts in the opposite direction to the motion of the body and can be calculated using the formula:
Frictional force = coefficient of friction × Normal force

3. Normal Force:
The normal force is the force exerted by the surface on the body perpendicular to the surface. In this case, since the body is on a horizontal surface, the normal force is equal to the weight of the body, which can be calculated using the formula:
Normal force = mass × gravitational acceleration

Analysis for the First Case (F newtons):
For the first case, the force applied is F newtons, and the acceleration is a1 = 5 m/s^2.
Using Newton's second law of motion:
F = ma1
F = 10 kg × 5 m/s^2
F = 50 N

The frictional force can be calculated using the formula:
Frictional force = coefficient of friction × Normal force

Since the body is on a horizontal surface, the normal force is equal to the weight of the body:
Normal force = mass × gravitational acceleration
Normal force = 10 kg × 9.8 m/s^2
Normal force = 98 N

Substituting the values into the formula:
Frictional force = coefficient of friction × 98 N

Now, we know that the acceleration is caused by the net force acting on the body:
Net force = Applied force - Frictional force

Substituting the values:
50 N - coefficient of friction × 98 N = 10 kg × 5 m/s^2

Simplifying the equation:
50 N - 98 N × coefficient of friction = 50 N

Rearranging the equation:
98 N × coefficient of friction = 0

Since the coefficient of friction cannot be negative, the only possible solution is:
coefficient of friction = 0

Analysis for the Second Case (2F newtons):
For the second case, the force applied is 2F newtons, and the acceleration is a2 = 18 m/s^2.
Using Newton's second law of motion:
2F = 10 kg × 18 m/s^2
2F = 180 N
F = 90 N

The frictional force can be calculated using the same formula as before:
Frictional force = coefficient of friction × Normal force

The normal force is still equal to the weight of the body:
Normal force = 10 kg × 9.8 m/s^2
Normal force = 98 N

Substituting the values into the