In Newtonian mechanics, the equation that relates force, mass, and acceleration is known as Newton's second law of motion. The basic idea behind the proof of this law is to consider the behavior of an object under the influence of a force and observe how its acceleration changes with respect to changes in mass.
Let's start with the assumption that force, mass, and acceleration are all vectors, meaning they have both magnitude and direction. The magnitude of force is measured in newtons (N), mass in kilograms (kg), and acceleration in meters per second squared (m/s²).
Newton's second law can be stated mathematically as:
F = m × a
where F is the force acting on an object, m is its mass, and a is its acceleration.
To understand the proof, we need to consider the concept of momentum. Momentum (p) is defined as the product of mass (m) and velocity (v):
p = m × v
According to Newton's first law of motion, an object in motion tends to stay in motion with the same velocity unless acted upon by an external force. This implies that the rate of change of momentum (dp/dt) of an object is equal to the force acting on it:
F = dp/dt
Now, we can differentiate the momentum equation with respect to time:
d(p)/dt = d(mv)/dt
Using the product rule of differentiation, we get:
d(p)/dt = m × dv/dt + v × dm/dt
Since mass (m) is assumed to be constant, dm/dt is zero. Additionally, dv/dt is the definition of acceleration (a). Therefore, the equation simplifies to:
d(p)/dt = m × a
This shows that the rate of change of momentum is equal to the force acting on the object.
Finally, by recognizing that the rate of change of momentum (dp/dt) is equivalent to force (F), we arrive at Newton's second law:
F = m × a
Thus, through the proof outlined above, we establish the relationship between force, mass, and acceleration according to Newtonian mechanics.