Understanding Force in Mechanics: Why Fmv/t is Incorrect and How Fma Resolves It

Introduction to Newton's Second Law of Motion:

Newton's Second Law of Motion is a cornerstone of classical mechanics, stating that the net force acting on an object is equal to the rate of change of its momentum. This law is often expressed as:

[F ma]

where F denotes the force applied to the object, m is the mass of the object, and a is the acceleration of the object.

Acceleration: A Fundamental Concept

Acceleration, denoted by a, is defined as the rate of change of velocity with respect to time. Mathematically, this is represented as:

[a frac{v - u}{t}]

Where v is the final velocity, u is the initial velocity, and t is the time taken.

In the context of an object starting from rest, the initial velocity u equals 0. Therefore:

[a frac{v}{t}]

Substituting the Expression for Acceleration

By substituting the expression for acceleration into Newton's Second Law:

[F m cdot frac{v}{t}]

We obtain the equation:

[F frac{mv}{t}]

Although this form of the equation is mathematically correct, it is not the standard form used in classical mechanics.

Why Fmv/t is Misleading

The equation (F frac{mv}{t}) can be misleading for several reasons:

Dimensional Consistency

The term (mv) represents momentum ((p)), which is defined as the product of mass and velocity. The dimensions of momentum are ([mass] times [velocity]), which is not equivalent to force.

Units

The unit of force is the Newton (N), defined as (text{kg} cdot text{m/s}^2). The unit of momentum, (text{kg} cdot text{m/s}), does not match the unit of force.

Thus, the standard form (F ma) is preferred because it maintains dimensional consistency and aligns with the physical units of force.

Conclusion: The Correct Formulation

The correct formulation of Newton's Second Law is:

[F ma]

Acceleration ((a)) is the key to understanding how force relates to mass and velocity over time. The expression (F frac{mv}{t}) can lead to misunderstandings and should be avoided in standard applications of classical mechanics.

References:

Bauer, H. W., Westwater, G. (2017). University Physics with Modern Physics. Pearson. Halliday, D., Resnick, R., Walker, J. (2014). Fundamentals of Physics. Wiley.