A car starts from rest to cover a distance x. The...

A car starts from rest to cover a distance $x$. The co-efficient of friction between the road and tyres is $\mu$. The minimum time in which the car can cover distance $x$ is proportional to
(A) $\mu$
(B) $\frac{1}{\sqrt{\mu}}$
(C) $\sqrt{\mu}$
(D) $\frac{1}{\mu}$

Key Concepts

Dimensional Analysis and Proportional Reasoning

Dimensional analysis involves using the dimensions of physical quantities to derive relationships between them. In problems involving motion, it assists in understanding how time, distance, acceleration, and friction interrelate. Proportional reasoning, an important mathematical tool, enables one to deduce how altering one quantity, such as the coefficient of friction, influences the time taken to cover a distance based on the established kinematic relationships.

Kinematics of Uniform Acceleration

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, acceleration, and time. When an object starts from rest and accelerates uniformly, these equations play a key role in determining the time taken to cover a given distance, establishing a relationship between time and acceleration. This is central to problems that involve calculating minimum times under limited acceleration conditions.

Friction and Coefficient of Friction

Friction, particularly static friction in this context, is the force that resists the relative motion of surfaces in contact. The coefficient of friction is a dimensionless value that represents the ratio of the maximum frictional force available to the normal force. It determines the maximum acceleration that can be achieved without slipping, which is critical when analyzing the motion of vehicles or any object accelerating on a surface.

Newton's Second Law of Motion

Newton's second law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This principle is used to determine the maximum acceleration a vehicle can achieve, given that the force is provided by friction. The frictional force, limited by the coefficient of friction, ensures that the resultant acceleration does not exceed a specific bound governed by the normal reaction and the coefficient.

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